That's how it works right now. Besides Trump, the US's last presidental impeachment was over a sex scandal. They "say" it was because he lied about it, but I think we know better at this point. It's just really hard to impeach because you need 66 or 67 of the Senate to agree on something, not the usual 51.

But yes, we'd need some truly neutral Ombudsmen to back up such a system. And they themselves would need to be accountable should they corrupt. I don't think it's impossible, but hard to do with the current power structures.