Did the shop kill my Daughter's Prius?

We just took our 2010 Prius Type 3 to a local "Made in Japan" shop and now, it is undrivable.
We took it in on Friday (9/3) and got it back about 5:30pm that day.
Almost immediately after leaving the shop's parking lot to get it home (about 2 miles), it was undrivable due to engine roughness (described below) and the initial roughness problem was worse than when we dropped it off.

The question is: Did they kill my daughter's Prius?
Please read on for more details.

Background:
For the last 2 weeks, on initial startup, there has been a bad vibration from the motor (shaking / rocking so hard there was some piece of sheet-metal rattling), but it would smooth out in less than a minute. It was bad enough that my daughter called me when it first happened. But it only happened when the engine was cold.
When I checked it, I heard the rattle, but there was also a subtle vibration in the steering wheel when at a stop if the IC engine was running.
I also noticed that was a pretty significant hesitation (hiccup) when I pressed the accelerator from a standing start. If I "babied it", it took longer to get through the hesitation than if I pressed harder.
But, there was never any smoke and the MPG seemed to be typical.
We were planning to have it looked at when she had some time next week.
Then, on Friday, the "Check Engine" light came on when she started it, so my daughter drove it over (about 20 miles) and she took my '16 Prius to work.

Friday:
We explained all this to the guy at Made In Japan, and he told us there was a Toyota mechanic working there that had 15 years at Toyota and they'd get it sorted.

When they called with their diagnosis, they said it needed new sparkplugs (4), Coil pack (1) and an EGR valve and they needed to clean all the EGR ports (remove the intake) in the head.
The code they pulled was a PO301 (Misfire on #1) and "egr ports plugged".
The mis-fire made sense due to the shaking.
I guess I can't comment on the EGR. But... maybe.
I agreed and they started working on it at about 1pm and called me at 5:30pmn to pick it up.
The guy said there was a little vibration, but it would clear up as I drove it. And, I should start using Chevron gas to keep the intake clean.
They did $1,500 work on the car between 1pm and when we picked it up.

Driving it:
The drive home was awful and the Check Engine came within one mile.
The problem that we took it in for was obviously worse. It never cleared up and it was present the whole drive whenever it was trying to idle.
I called the shop when I got home and they didn't answer, so I left a VM.
They will not be open until Tuesday (9/7), the day after "Labor Day" (3 day weekend).
I didn't drive it or start it again.

Saturday:
After the drive home, I let it sit in the driveway overnight. We went out to start it the next morning and I was taking a video of the engine (hood opened) while my daughter started it and let it run.
Watching the shaking was dramatic and the noise was pretty bad. Just looking at it you could tell it was in trouble.
After about 30 seconds, the engine tone changed a little and it smoothed out (stopped shaking and the rattling died down) to what I considered to be about the normal sound and idle speed of the motor.
And, at that moment I started smelling something burning or really hot (I would compare the smell to a smoking clutchplate or hot brake pad smell) and there was a huge, thick, white cloud coming out of the tailpipe (like it was vaping). I hadn't seen the smoke because I was looking at the motor.
As soon as I smelled it, I told her to shut it off and noticed the smoke trailing off downwind.
In the video, I tried to catch the cloud as it blew away, but when I looked at the tailpipe, there was a residual vapor slowing floating out of the tailpipe for about 1 min.
I haven't try to start it again.
Its sitting in the driveway waiting until Tuesday when I plan to have it towed back to Made In Japan for them to "do something"

I bought this Prius new in 2010 and gave it to my daughter in '16 with about 95k miles. It now has 160k.
We have always taken it to the dealer for routine maintenance and never had any issues.
Outside of the suggested maintenance in the owner's manual which has been at done at the suggested intervals, the only additional work (literally) was new tires (when needed) and replacing a dead 12v battery about 6 months ago when it died in a parking lot (by the dealer).
Its been an awesome car and I'd hate to think that the first time I didn't take it to the dealer, it was killed.

I think my biggest concern is that the catalytic converter is burned out due to the misfire that has been present for the last 2 weeks. I've never smelled a burning catalytic converter, but does it smell like a burning clutchplate and produce thick white smoke?
Also, is there any chance that if the cat was getting raw fuel from the #1 misfire for a few hundred miles, it would clean itself out and be okay when the problem gets fixed?

I could understand why they would zero in on the misfire in the #1, but why would they change the EGR valve?

Thanks

 

White vapor in the exhaust = water vapor = blown head gasket?
I've blown head gaskets on a TR6 and one bank of a 389. And I've burned a lot of clutch disks.
But, I don't recall the white vapor being this thick or the smell of burning "friction material"
And, the white vapor only showed on Saturday morning - not over the course of the 2 weeks it had (what I assume) was the misfire.
I will search for "2010 head gasket".
But wouldn't I expect to see water in the oil or oil in the coolant?

And, a head gasket would be the final nail in the coffin for this 2010.

 

As is true for any shop.

The EGR valve ($319) is somewhat visible on the driver's side on top of the motor. It looks new (at least, it has the "new plastic" shine to it).
I can't confirm the new plugs ($19 each x 4), the new coil pack ($113 x 1) or the EGR port's cleaning.

I would "assume" that if they thought it was a bad head gasket, they would have had visions of a new boat, their kid's college tuition or a tropical vacation on their minds and told me about that instead.

[EDIT]
Searched and found the "2010 Head Gasket" threads - there are quite a few.
But, a lot of the more technical replies include suggestions about replacing the EGR valve and cleaning as (more of less) a 1st step.
If they cleaned and replaced the EGR valve, they may have looked at the oil levels (normal) and the lack of smoke / vapor and thought that would be the most likely (least expensive) place to start.
I'll also ask if they did a carbon monoxide test on the coolant.

 

Thanks for all these comments.
Updates (9/10):
The car is now starting and driving more normally.
It sat in the driveway last night, and this morning the startup was drama-free and nothing unusual in terms of sounds and no visible smoke.
My daughter then drove over the hills and through the woods to her job (about 1 hour in stop & go traffic and up and down a steep grade) with no problems.

Work:
The shop called me at 8am Tuesday as soon as they listened to my VM from Friday.
They actually sent a guy over Tuesday morning (around 9am) to drive it back to the shop.
They had it for Tuesday, Wednesday and Thursday saying they were doing "diagnostics" (as in: driving it, letting it cool off, watching for smoke - etc). I would guess they spent about 4 or 5 hours without billing me.

During that time, they drained a couple quarts from the trans (using the drain plug - not a flush) because they also noticed the weird hiccup when pulling away from a stop sign, and said the oil was definitely toast (as in "burned"). I had never said anything to them about my suspicions about the trans. They recommended a flush.
They also said they suspected a "small" head gasket leak, but shouldn't do a head gasket replacement due to the mileage (160k). They suggested a "stop leak" head gasket sealer instead.

They added something called "Blue Devil Stop Leak" ($45), did the Trans Flush (8qts) ($210) and cleared the Check Engine light (PO301 & P0302 - misfire in #1 & #2).
Then, they drove it for an hour (as instructed on the stop leak) and then drove it around town (stop & go) for another hour. Then parked it.

When I picked it up last night (Thursday), it was significantly better overall.

They tell me this is about as good as it will get (in terms of the IC engine idling and startup vibrations), but the car is on life-support and we should start looking for a new car (they don't know how long before the stop leak "wears out" and the trans may need more than just a flush).
Driving it around last night, it pretty much felt "normal" and the Check Engine light didn't come back.
My daughter reported that the drive to work felt "normal" and the Check Engine light is still off (yea).

Notes:
It's been at least 4 years since I've had to drive this car. I gave it to my daughter in '16 and there was never a need. It was her "first car".
The only reason I drove it recently, was because she told me about the horrible rattling on startup (about 3 weeks ago).
While I didn't get to experience the horrible ratting on that drive, that's when I noticed what felt like a misfire vibration in the steering wheel whenever the IC engine was running at an idle (like a stop-light or stop sign), it pulled to the right on a flat road and had a weird hiccup (press the gas and the motor started spinning up but it wouldn't start moving) when pulling away from a stop-sign, driveway or a red-light.
After that, I drove it again to get the frontend aligned (I check the tire wear and pressure to rule that out as the cause) and told my daughter we were going to have to get some serious service done to it before long.
About 5 days later, the check engine light came on and this story begins.

In general, there have never been any significate (in terms of cost) maintenance done to it. As I said, other than normal dealer maintenance (based on the Owner's Manual) that was done at the recommended mileage internals, nothing else was done (no EGR, No Trans Flush, No Oil Flush, No Stark Plugs, No Coil Packs - etc). If it wasn't in the Owner's Manual, it wasn't done.
There is no doubt I'd have paid more attention to the car if I'd checked this site back in '12, '13 or '14. I would would have been more proactive with things like EGR and water pumps and trans fluid.
I'll start looking at the PriusChat section dealing with the '16 model today and mapping out a plan for that one.

Side Note:
RJParker:

Gasket Masters.
Holy Cr@p. If they do what they say at the price they quoted, it has to be the best value I've heard of in quite a while.
I emailed them Monday night, and they called me back Tuesday morning.
Their shop is in Manteca, but they will come to me to do their work for a few hundred added to the price.
They said they can do a head gasket change or change the motor in about 3 hours. And do in my driveway. They will show up with a spare engine in the truck, and decide whether to change the gasket or the engine and then get'r done.
The thing that blew me away was the price. Even with the surcharge added for travel, the cost of the head gasket replacement was much less than I expected, and to replace the motor was only a few hundred more.
If the problem in my daughter's car comes back, its going to be hard not to decide to just drop a replacement engine in it instead of getting a new car.

Thanks