Traction control and AWD are two different things. They could use the same inputs, but turning off traction control leaves the AWD on. The concern is the different sized wheels spinning at different rates could stress and damage the differentials. This could also happen ith a 2WD drive system. The instructions to drive slow with a spare isn't just for the tire.
According to Toyota the inflator can be used without the sealant (1:02 min into the video). However, they do not instruct how... (Toyota How To: Tire Puncture Repair Kit | Toyota USA)
Consider too, why are you going to use the inflator. It will likely “total” the tire, whereas a plug repair could fix the leak and it’ll more’n last till the tire’s in need of replacement.
I once had a 4WD where that was a definite concern, as it had no center differential. There were differentials both front and rear to allow turning corners, but when 4WD was engaged at the transfer case, the front and rear driveshafts were expected to turn the same speed. In snow or sand it would be possible for something to just slip a little, but on dry pavement any slight difference between the front and rear effective diameters would work kinda like a Chinese windlass and try to twist up the driveshaft or break the transfer case. (Or, less annoying, could just get the transfer case so tightened up it couldn't be shifted out of 4WD before jacking a corner up to let the stress off.) That kind of thing, at least, isn't a worry in a system that has a center differential (or, like a Prius, just has a whole separate electric motor for the rear). I don't think an open differential suffers much from even a long sustained speed difference between its outputs. A limited-slip differential might. But we generally just have open differentials now, with traction control able to brake individual wheels to take care of limiting slip. So maybe in the modern setup there really isn't much concern other than getting traction control warnings about a wheel-speed difference.
Just don't put the sealant bottle between the pump and the hose. The connectors are sensibly the same. Did this myself to reinflate a screw puncture, got me to the repair place.
Don't forget VSC and ABS. I suspect the car will just shut those systems down, leaving the driver without them to help. Can't totally discount it leading to drivability issues when those systems get mismatched wheel speed readings.
Yeah, the same ECU does all those things, and will light warning lights if it disapproves of the wheel speed readings.