I'm pissed. Spent way too much time unable to explain it to coworkers, thinking I just didn't understand Heroku's platform and that it was my fault.
Turns out, I didn't understand it, because Heroku never thought to clearly mention something that's pretty important.
Easiest fix: moving to EC2 next week. I've wanted to ever since our issues became evident but it's hard to make a good argument from handwaving about 'problems'.