going to dilute and cheapen any legitimate criticism of him because people switch off
I don't think this is accurate. I had that experience with all of the accusations surrounding Hillary, wild stuff like how she stole a bunch of furniture from the state department, or how she murdered the cat of one of her former aids, and I did as you said and just stopped paying attention. That doesn't seem to be how the majority of people handled it though. It seems as though the nonstop accusations did have a very significant impact on how people perceived her, though none of those things ever even got as far as a trial, let alone a conviction.
There's an old rule that if you repeat something often enough people will start to believe it, no matter what it is. Let me remind you that the difference between a religion, which we generally venerate, and a cult, which we generally denigrate, is only age - how long and how often beliefs have been repeated is what really matters.
Put not your trust in money, but put your money in trust.