Correct. The letter is not a dismissal of the merits or an endorsement of the merits.
If I can, perhaps I can sort of reframe your complaint as I understand it? Tell me if I’m close:
The gym barred you after the parking lot incident. However, that was a pretext, an excuse. The real, or at least major contributing reason was your on-going complaint about the gym’s failure to provide accessible equipment, specifically a lat pull-down machine.