Oooooh, I really like shifting Claims/Foundations over to the Downtime phase. It feels like that's the mechanic that gets the least amount of engagement from people I've played with and if I had to guess I'd say it's because a) most of my fellow players are new to the system, and b) there's already a lot going on in the Free Play and Score phases!
Also with you on simplifying healing. I think a hidden design principle in BitD is that players ought to have more than one character ready to go. Get injured? play a different character until this one is healed up. Too much heat? send a character to prison to cool things off. Overindulge? might need to play another character for a bit! So I guess Heat as-written makes sense in that context but I'm not a big fan of it.