I wouldn't say core problem but it's definitely a critical component missing from any ernest attempt at actual democracy.

The House and Senate have complete control over their members.

Each House may determine the Rules of its Proceedings, punish its Members for disorderly Behaviour, and, with the Concurrence of two thirds, expel a Member.

It would take 67 votes in the Senate, but they absolutely could expel him. That won’t happen because it would require getting now 12-14 republicans on their side to kick him out, and he’s useful to republicans.

Not for his voting patterns

I think that's the only real solution to the problem that's laid out in the current legal framework. It's historically pretty difficult to pull off though.