So, to update: I got the game working shortly after posting this issue - had to delete the steam copy and download/reinstall through the origin app so that origin would recognize the steam purchase. A bit of a hassle, but ultimately this worked to resolve my issue. At least until the recent origin app update. Now for some reason it fails to recognize the game is already installed. I went to click the download button to reinstall heat through the origin app (as this resolved my issue once before), only this time I get the same error: "sorry, it looks like need for speed heat deluxe edition isn't installed." So now, not only can I not play it, I can't even download it again to fix it the way I did the first time.
And yes, I've cleared my cache. I've restarted the pc, the app, the launcher; I've tried starting the game through steam (no controller plugged in), through origin, from the shortcut on my desktop - they all bring me back to origin's inability to play in the sandbox with others.
It's sad that a $75 purchase can't be enjoyed. On a slightly related note, how does one request a refund? In my case, probably through steam... but with the way things are going, they won't recognize my origin game was ever purchased...