Before joining her first round, Alice is expected to only have non-private coins
This expectation is what allows Alice to detect the Sybil attack because her other private UTXO will be rejected.
By registering 2 inputs which are tagged from blockchain analysis as belonging to the same person, it tests absolutely nothing. The coordinator can still attack, since they know with certainty that these 2 inputs belong to the same person.
Knowing that 2 inputs belong to the same person still doesn't allow the attack to go undetected because that person could have a third input.