Just going to add my experience here as I just transferred my car I purchased from a 3rd party dealer into my Tesla account. For the transfer experience itself see:
Trying to claim ownership on Tesla website, but keeps saying “unable to process, try again later”
Dealer had purchased vehicle from a private 3rd party and not from auction. Car had FSD included in package (showing on screen) and things like NOA was there and working, also had button to request FSD beta.
Dealer had called Tesla for some other information and also asked them about FSD. Tesla confirmed that FSD (or EAP, ...) is only removed if the car passes back through Tesla.
After transfer of the car into my account, which was pretty easy and fast (less then 10 minutes, see other post I linked above) the car kept FSD. The car definitely received a config update. Notifications on screen showed a config update was done and the car had rebooted. This was part of the transfer into my account.
The issue I can also see with some is when the car for example goes back to Tesla, sold on auction to a dealer, then transferred/sold to another dealer, and then sold to a private party.
I also think that what might happen is that it is only when the car is transferred then into an account it receives the config update. So in my example, if neither dealer added the car into their own Tesla account then the config update hasn't happened yet. Then when the car is finally transferred into someone's account the config update occurs resulting in FSD or so being lost as it had passed Tesla.
Best way to prevent this would be by making absolutely sure dealer bought car from 3rd party (and check Carfax or so to ensure person had the car in ownership for a while as they clearly would then have had it in their account). If dealer bought car on auction then make sure dealer has the car transferred into their own account since if my theory is correct the config update would have occurred and if it had passed back through Tesla the options would then be removed.
Otherwise, might be able to call Tesla as I would think they might be able to check based on VIN if the car had gone back through Tesla and if FSD or so would thus get removed or not.