A professional tennis player is someone who gets paid to play tennis. The definition cannot vary.
But many players get paid to play tennis at some stages in their career, but in many cases that is never enough to support him. Which I why I doubted that the OP meant that vast definition of professional player, otherwise the answer is pretty straight forward: a very small %. he said: "But what about those great players who failed to win one"... clearly he does not mean all professional players.
Also what tournaments does he mean? ATP tournaments?