The p2p network can't verify them if it doesn't have the full chain, which is exactly what we were talking about: lightweight nodes that run an ARM thumb or something. You have a bootstrapping problem.
Yes archival nodes would (or at least could) verify the whole thing, but there is nothing forcing them to, or at present not even any incentive to run an archival node at all. What happens if no one realizes that the last archival node shut down, or is malicious?
To be clear, I'm not arguing "against" UXTO commitments, just pointing out that there is a reduction in trustlessness if everyone isn't verifying everything, and added risks when the incentives to verify are aligned with letting "someone else" do it. These can probably be fixed, or at least controlled, I agree with what you said about the third party trust creeping in regardless.
The thing to remember is you don't need to hold the full chain to participate as a full node. That is why pruning works.
Once you have a valid node state as of xxx block, you only need the header chain and the UTXO set to FULLY validate the next block. Assuming your node's state is correct, you do not need the full history to validate the next block. Again this is why pruning works.
A UTXO set offers a faster method to reach a valid state as of xxx block.
Today the only method to reach a valid state is to download & validate the complete history. But once your node has done so, your node can delete that history and just save the UTXO set, and still validate all future transactions and blocks. This is exactly what nodes that prune do. But this method is very inefficient, it is not a problem today, but it will be if bitcoin succeeds.
As another example, with pruning it is possible that all nodes on the P2P network prune their history, and NO nodes on the network have a complete history individually. But each one would still be able to independently validate transactions and blocks. This is because they still have the valid UTXO set, the pruned history is not necessary.