How does competitive chess affect players' mental health? I suppose that participation in international open tournaments has a positive mental effect on most amateur players. As for the professionals involved, if they have other sources of income and also don't suffer from untreated issues then I can't see any mental health risk. Concerning international closed tournaments, there players have already been preselected both by their coaches and the organisers, thus I assume most of them should be well-adjusted.
if there are certain talented players who should not become competitively active cause of their mental health status

On the contrary, there are many chess players, successful and otherwise, who get relief from their mental problems by playing chess. Fischer and Bloodgood are probably at the extreme end of mental disturbance but there are many at all the levels below that.
Victor Kortchnoi, in his famous world championship match with Karpov, is another famous example at a lower level of mental disturbance.

others who could become successful only under the supervision of a mental health practitioner

Again, I think you have it the wrong way round. Without their mental health problems many strong GMs wouldn't be that strong.