They have a big problem in their design where currently it works exactly as you described, and at "some unknown point in the future" when the database of all transactions EVER in the history of bitcoin gets too big for each person to have to have a copy of in order to add another transaction, they'll "figure out a way" to make it unnecessary.
Can't they somehow publish the current state of things, let anyone who wants to verify it from the public record building a web of trust, then go from there forward?
They have a big problem in their design where currently it works exactly as you described, and at "some unknown point in the future" when the database of all transactions EVER in the history of bitcoin gets too big for each person to have to have a copy of in order to add another transaction, they'll "figure out a way" to make it unnecessary.