To play devil's advocate. All mental exertions are physical exertions unless you subscribe to some theory of mind-body dualism. So given that mental activities and physical activities are to a large degree interrelated, quite possibly co-dependent if not co-terminus, then yes at any point you do a mental activity you are necessarily doing a physical activity. Moreover, all physical activity takes mental activity as it roots (with the exception of reflex systems). Partnered with the skill and competitiveness of chess there is an argument to be made for it as a sport.

As for synchro, my sister used to do it. It takes a lot to be good at it. Just because it looks easy does not make it easy. The fact that it looks so easy probably is indicative of a lot of work and training.

More to the point, do I actually care if chess is a sport or not? NO.