Why I've lost faith in p values

There has been a lot written over the past decade (and even longer) about problems associated with null hypothesis statistical testing (NHST) and p values. Personally, I have found most of these arguments unconvincing. However, one of the problems with p values has been gnawing at me for the past couple years, and it has finally gotten to the point that I'm thinking about abandoning p values. Note: this has nothing to do with p-hacking (which is a huge but separate issue). Here's the problem in a nutshell: If you run 1000 experiments over the course of your career, and you get a significant effect (p .05) in 95 of those experiments, you might expect that 5% of these 95 significant effects would be false positives.