This hypothesis is strongly challenged by the fact that crime has also dropped in other developed countries that do not have mass incarceration.
Anyway - no birth control method (except abstaining) is 100%, even when you stack multiple different ways it can happen.
And some methods of birth control - like pill for woman and vasectomy for guys - can mess you up long term.
Women describe getting off the pill like completely changing their taste/smell senses, even brain/thinking ...etc (which sounds similar to what happens in/after pregnancy). When they give it to teenage girls (to "fix" their irregular cycle) they often gain weight..
And while vasectomy could be reversed, it's not guaranteed it would work.
Also worth noting that there are about a million birth control pills on the market with different side effects... and different levels of efficacy. Progestin-only pills are popular because people tend to tolerate them well, but they're notably less effective than pills with estradiol in them. Sometimes that's a worthwhile trade-off!