This idea just flash in my brain. How about let's flat cars have ability to display a block or an item on it...
(Like item frame from vanillia minecraft and tables from BiblioCraft mod)

Here's my suggestion：Make a new flat car instead of have a seat on it, When you right-click on it and it have only one slot on it's gui (Like table's gui from BiblioCraft mod), When you put a block or an item on it, and it will display on the flat car, like you put it on to the item frame.

P.S: Sorry for my poor grammar, My English is not good.

Why can't it be one vehicle which allows you to "mount the vehicle" when you open its UI?

I don't understand what you're saying @Bendragon1337 you mean like make the mount a UI button rather than a right click action?

Yeah, Like you can with Steve's carts.

When you have a seat and side chests, you can choose to insert stuff into an inventory, it also allows you to mount the vehicle by clicking on a button.

that is an interesting option.
theoretically it might even simplify the mounting code in the rewrite. I'll look into it.