It's a statute. You have to apply it. That invokes a judicial proceeding. The conviction - not just the accusation - has to be produced, or that particular statute can't be applied, by definition, because that statute references a conviction in its own definition of itself. Hillary's defense would be that simple: she rejects Comey's "opinion."
You are absolutely correct. She has to be indicted or presented with a criminal information first, and found guilty by plea or trial, and convicted in a United States District Court before the statute means anything. The burden of proof doesn't change. Law enforcement can accuse or state that laws were broken all of the time, but the individual must be convicted in court before any of the statutory provisions apply. End of discussion.
Also, the qualifications for President are laid out in the Constitution. Wouldn’t adding additional qualifications by statute, rather than by constitutional amendment, be unconstitutional?