Except of course the time the OP arrived at the desk which will then be COs word against his and I suspect the claim will be that he arrived the gate within the 15 min period. As OP said the GA were using delay tactics to ensure they got him to the 15 mins. I once had a similar experience on VS where a checkin agent told me a flight was closed 4 mins before the hour mark and by the time the arguments etc finished was truly beyond the hour. The supervisor apologized and upgraded me to UC on the next flight.

I have seen that a lot of these airline agents are laws unto themselves and when it comes down to it in event of complaints it will be their word against theirs. I bet they had given out the OPs seat and were trying to save face. Did both Y and F check in full for the flight?