Yeah, you're right about that. I'll admit this topic sent me down a bit of a rabbit hole on how this could be accomplished.
An alternative scheme could be a Vickery Clarke Groves or generalized second price auction. These bidding schemes are used to price online adverts. Once the auction closes, let people choose their seats in order of who paid the most. This let's you place one bid, and individually prices each seat, but the downside is you don't know how good a seat your bid will get you. Still, would probably be easier to understand. If you bid more, you get first pick.