No you don't need to track UTXO's because UTXO's are stupid. Them and blockchain are the constraint of the whole system.
The distributed ledger is all that matters. You don't need to know "who sent what". You only need to know "who owns what". That's it. The ledger.
The logic is simple: Has key? Has balance? Send approved.
This is kind of the idea pushed even farther with MimbleWimble on things like Grin, Beam, and LitcoinMWEB
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