If you kept paying off corrupt doctors you set yourself up to be blackmailed for even more money and eventually the truth would leak out anyway. You'd have to pay big money and even then they could leak the information some years into the future.
What would you do then as a player? Kill them? This is the real world not a fantasy movie.
Yes the truth comes out, like the fact that many tennis players have been treated by Luis Garcia del Moral. One of Armstrongs former doping doctors.