My dealings with insurance claims have been all over the place. Some go well and some not so much. My DW has sold me all of our current policies and even with having an in with my agent I still have not gotten a favorable response for why they won't cover my drivers side front dual pane window (another rant for another day).
I would have also done exactly what you did. In fact I may have already moved the coach to a safer location prior to contacting GEICO roadside. I always have taken the stance that if it worked prior to the event and doesn't work after the event than the event must have been the cause. I get the whole wear and tear and normal maintenance but in a case like this it is pretty straight forward that a tire failure was the leading cause to the needed repairs. Any damage that would have been caused by moving the coach to a safer location still is a result of the initial tire failure. Complete negligence in maintaining the tire would be the only reason for denying the claim which I highly doubt is the case here.