I'm not sure about that - the NASA commentator says the LES tower was jettisoned pretty much at the time the booster appears to have failed. From her later comment, she doesn't seem to have seen that a failure happened until some time afterwards, even though the Russian's are clearly saying a failure happened, so I think she's reading from a script of what should be happening at that moment. So I would not treat her saying the LES tower jettison occurred as being certain.
I agree with your interpretation of the tape audio, but the 'ballistic trajectory' statements from Roscosmos/NASA tell us that it was not rocket-powered. Indications are that the anomaly happened at booster sep, which is a few seconds after LES jettison.
That page says LES can be jettisoned on either side of separation depending on flight profile.
The tower is not the only component of the Soyuz LES - after the extra thrust is needed, the tower is jettisoned and the LES relies on the engines in a "shroud" around the capsules.