I think the way to run a heist without flashbacks would be to run it as linear as possible. You (the DM) would have to prep the place to be heisted, then y'all would have to play out all the info-gathering, planning, and anything else (such as social-engineering) that needed done before the heist, then y'all would have to play out the heist. You (the GM) would have to have the players' trust that you'd run all of that in good faith.Maybe, but that is specifically what I don't like about BitD. I would be interested in heist/game mechanics that don't involve flashbacks.
I think 5E could work pretty well mostly as-is for this. I think it's an entirely different playstyle than BitD is aiming for, and I'm not sure how much anything from it (other than maybe clocks) would be applicable. It's plausible, as @tetrasodium seems to imply, that maintaining believability will be difficult if there's asymmetry of real-world security expertise between the players and the DM. You're a far better judge of whether that's likely to be a problem at your table than anyone else in this thread is likely to be. (IMO, if no one is an expert in the field, it probably won't be a problem.)