Again, the way I see it, public tennis courts are "first come, first served", regardless of whether they have a maximum time limit or not, so long as you are actually USING the courts (you can't claim "phantom reservations" unless it's expressly allowed).
When the OP arrived, the court in question was NOT being used, despite what the tennis coach claimed. So the OP had every right to use that court.
In fact, I think the tennis coach only irresponsibly escalated the confrontational situation by sending CHILDREN onto the court while the OP was using it. The tennis coach was wrong on 2 counts: first, to escalate the situation; second, to use children to escalate the situation.
This is why I think the OP did the right thing to not back down by surrendering use of the tennis court.
If it were my children at the tennis clinic, I would have been mad at the tennis coach for recklessly involving children in a needless escalation of "hostilities" and would have reported the tennis coach to authorities and his manager (if any).