It's not just the athletes in whose interests it is to maintain this polite fiction.
How I would see full disclosure going would be something like this:
In something like tennis, the top 100, say, or however many it is that make a big living out of the game, which is about the top 100, must submit details of every prescription drug taken along with an explanation. Every 6 months, say, they must submit a log of the drugs taken. They are then liable not just to random drug tests but to random honesty tests - where a second opinion is sought and hospital records along with blood tests, etc, must be submitted to support the claim of the particular condition. For instance, clearly a rash of heart conditions hasn't broken out among athletes even if each individual has a doctor's note - individual doctors can be bought off.
A big intrusion into privacy? Yes, but the current regime is already that, and you're only expecting it from those to whom you are paying out big amounts of prize money.
Either that, or you have to allow everyone to take anything they like as long as it isn't banned - which is pretty much the status quo - which would be dangerous as it might almost force everyone to take the latest dangerous thing to stay competitive.
Otherwise, you can't get honesty. Would something like that be workable? The default position really should be that if you're not ill, you shouldn't be taking prescription drugs - most ordinary people manage to do that, and so should athletes. If that means you can't train quite as hard, well, that's the whole point.