Based on past experience, register thumbprint doesn't mean much if they didn't get your car grant(+key la). it's just to confirm you are interested to trade in w/ the dealer and stored an ownership transfer draft. A bit sus though if you agreed to let them process & transfer money after you pass them car key + grant, then you have to wait I guess.

Like I said is my past 2 cars trade in experience, maybe different practices. I did thumbprint with dealer A as suggested but I ended up with another dealer B who offered higher, they won't simply go through that process unless car, grant + key is with them. Same with recent one, I agreed to thumbprint with A as offer is good yet awaiting answer from dealer B, no ownership transfer happen until I confirmed I go with A a week later and passing all items to them. It's a bit troublesome though I got to compare few options. 1st one money is in on same or a day late. 2nd one is partner w/ car dealer so they handle themselves.