It is absolutely true. It is conspicuously posted in bold on the UA website (see below). The fact that in your particular case you were not off-loaded means nothing. If there is space and you arrive at the gate before the flight is closed, UA will board you (as will AA & DL). But, at T-15, if the flight is overbooked and at risk of oversale or there are simply standbys waiting for seats, UA can offload you.

Here, OP was clearly not at the gate by T-15 as he was still at his arrival gate.

Frustrating? Yes. But, within UA's guidelines and done to keep the system operating on schedule.

As an aside, not worth arguing with the arrivals GA. If things are tight, make a run for the departure gate. There is always the chance of a last second delay and always the chance that someone offloads themselves or noshows. The worst that happens is that you have made a run for it and failed.

"Boarding ends 15 minutes before scheduled departure for flights within the U.S."