Update here: since changing the EGR, I haven't had a single issue!

Lesson: if you're having throttle issues (I.e. Hard idle, shaking after idling for a while) and you've already cleaned the MAF, filter, and throttle body, go for the EGR valve next!

 

You're going to need a new EGR. The MAP sensor is built in to it, and you're looking at a sensor failure, not just it being dirty/gummed up.