The wrong statement is saying P(no real effect) < 5%

The correct statement is saying P(saw these results | no real effect) < 5%

Consider two extremes, for the same 5% threshold:

1) All of their ideas for experiments are idiotic. Every single experiment is for something that simply would never work in real life. 5% of those experiments pass the threshold and 0% of them are valid ideas.

2) All of their ideas are brilliant. Every single experiment is for something that is a perfect way to capture user needs and get them to pay more money. 100% of those experiments pass the threshold and 100% of them are valid ideas.

(P scores don't actually tell you how many VALID experiments will fail, so let's just say they all pass).

This is so incredibly common in forensics that it's called the "prosecutor's fallacy."