Everyone, including criminals, should be able to vote or run for presidency. In the end, what’s a crime and what’s not is decided by the law, which is made by the politicians. What’s to stop a president from making something illegal for the sole purpose of making their opponents criminals so they are guaranteed a win next election?
If a president can do that, there's something wrong with the way your government is set up. I disagree, it affects personality cults the most and if it's about the issues that you care about, there should be no shortage of candidates. This is someone you are electing to lead your country for four years. I would concede that it does depend on the severity of the crime.
Oh, there is quite a bit wrong with the way our government is set up. We each break multiple laws per day and it's entirely a matter of which ones they want to prosecute. They've already targeted indigenous and black people with overly broad laws, and I wouldn't put it past them to go after Muslims next.
There is no direct line out of this mess; we'll have to fix multiple things in parallel.