If the definition of "self-proclaimed" includes agreeing with popular proclamation, who is ever proclaimed but not self-proclaimed? You've rendered the "self-" prefix redundant, in which case why include it?

Winning an election is not a proclamation or any other kind of indication that somebody is an expert in law, governing, or anything else except winning elections. I can't work out how you've made that leap.

Election to office is an indication that voters generally agree with your claims of expertise, and thus are proclaiming you an expert.

Voters who want someone in office who is not an expert at governing vote for someone who doesn't claim to be an expert.

It's no such thing.