I’ve often booked and sat in the priority seats and have never had any issues with passengers or staff.
The train managers do walk through the train and always seem happy to answer queries or help, but I’ve never had any reason to engage with them myself and they’ve not had to engage with me either.
Eurostars can also end up fully reserved, and in any case they don’t display which seats are reserved and which aren’t, so I imagine it’d be a bit of a hassle for a train manager to demand you to move as they’d have to locate an unreserved replacement seat for you. Whilst they probably do have a list of reservations or something, it’d probably be a bit of a time consuming task.
Perhaps if a passenger who’d booked a non priority seat boarded and realised they actually did need a priority seat, a train manager might ask if people in nearby priority seats were willing/able to swap? But even then I can’t see ‘proof’ really coming into the equation.
I think there are also quite a few priority seats, 4 per standard class carriage on 374s, which is more than any other class of train I can think of. So as far as Eurostar are concerned I imagine they consider them to be a normal part of the trains seating capacity.
I hope this helps
