So yeah, that's great. To bootstrap you ask nearby nodes what their max block is and their current UxO tree and for the block-header-history. It doesn't matter if the one you ask is a few blocks behind, since once you verify you can just append new blocks to the end. If you end up happening to bootstrap on a fork you just grab the UxO tree from the split and append the blocks on the longer chain.
The idea would be to have a root per block. However, it looks like it actually aux merkle tree for all transactions.
So, if you provided all the transactions for a block, you could prove that the update was correct. It requires more info per transaction though. You have to provide the path through the merkle tree.
This is required to make sure that no new money is created.