My 2010 does the start up rattle too once in a while. On mine which is a Gen 10 Corolla the VVT gear is bad. Really common. Youtube it. Lots of videos about that gear and the rattle. Was not aware the newer models are still doing it. I'm running 5-30 M1 oil some posts say running thinner 20 weight oil seems to help so I bought some 5-20 M1 for next time. Join Toyota Nation Corolla Forum. Great forum.
My view has always been, if it ain't broke, don't fix it lol. The noise has never got worse, and only upon startup.
That attitude would be fatal in the aircraft and especially in the airline industry. Deferred maintenance never works out. Small defects never get better by themselves.