Oh, I took the information about “exactly 4 wheels” literally and didn't even try building bigger single wheels out of multiple wheel components. Maybe it's just me but making it clear in the instruction for building the vehicle could prevent such misunderstanding.
Regarding not being able to move — I think I had a look at UI elements before giving up and quitting the game in this situation, but probably I indeed just omitted the inventory limit (I was aware of its existence and how trunk works beforehand).