I'm wondering if there is some confusion here.
Yes, someone needs to have a lookup table from prevouts to TXO tree positions. But because an insertion-ordered TXO tree does not rebalance, that table can be maintained by wallets or service providers for just their own coins, instead of by every full node and miner individually for everyone's coins.
In the simplest committed TXO model, full nodes simply maintain the TXO root hash, and every transaction/block comes with a proof that its inputs are in the TXO tree, and the necessary information to update the root after spending the inputs and adding the outputs.
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Pieter