If people who break laws can't vote, and the government decides what the law is and appoints the judges who enforce those laws, then the government currently in power can decide who gets to vote. Obviously there's an incentive there to make laws that disproportionately affect those who weren't going to vote for you, and thereby remove most of your opposition's votes. That way lies dictatorship.

It also makes it hard to change bad laws. For a random example, there used to be laws against homosexuality. How do you think LGBT acceptance in law would be doing if anyone who was openly gay or trans lost their right to vote? How do you improve access to abortion if anyone who has an abortion, provides an abortion, teaches young people about abortion, or seeks information about abortions becomes unable to vote? How do you change any unjust law if the only people who can vote are those who are unaffected - or indeed, those who benefit from the status quo?