I think the bigger problem with professional politicians is that they completely lose touch of why they are there and who they represent to begin with.
Case and point: Cali Gov. Gavin Newsom holding a birthday party at the French Laundry restaurant while telling everyone to cancel Thanksgiving because of COVID. Now I ain't the moralizing type who jumps on personal decisions to call out hypocrites the way, say, Fox News does. (Which, IMO, seems like the Republican version of cancel culture, IMO.) But really, dude? Are you that dense and stupid as to not think any of that through? How it looks to the message you send to the sense you are in this with anybody else?
The challenge with professional politicians is that getting reelected is their primary goal. I can't think of many politicians who would rather do the right thing and not get reelected, which is the completely wrong set of priorities we need.
As for the problems with the lottery system? 1. It's not democracy (nobody really gets to vote), 2. like jury duty a lot of people really have no business in a position of authority (and jury selection criteria applied to politicians is a whole other can of worms), etc...