Let me start by prefacing that, to date, this has been largely a hypothetical discussion.
I haven't yet seen evidence that MTG has done anything wrong, aside from having wacky politics. I guess it's possible that she has, sure. But the fact remains that, whether she has done a thing or not, either way the left would allege that she is guilty of every sort of moral failing under the sun. It doesn't matter which conservative or right-leaning politician Christians vote for: this will hold true no matter what. For obvious political reasons there's bottomless incentive to either point out whatever's there or, if nothing at all, manufacture stuff from thin air. This might as well be an ironclad law of American politics.
So far as avoiding support for a politician who the left will accuse of wrongdoings and immorality, the only way for this to happen is to either abstain from politics or capitulate all of their values and vote Democrat. A famous anecdote has it that even Billy Graham had an aide who would enter rooms ahead of him because journalists would try to ambush him with promiscuous women accompanied by a photographer, so it's probably not a stretch to say that Jesus himself would be made into Satan if he had an (R) next to his name.
That being said, we are supposing here for the sake of argument that a given politician is in fact immoral or otherwise unchristian in their private life.
Well that would suggest that a Christian can’t vote for a Jew or a Muslim, etc
You're saying that Christians should refuse to vote for a person who doesn't reflect Christian values.
If so, that necessarily excludes anyone who isn't a Christian, as non-Christians reject by default the most important Christian value: turning to God in repentance and accepting the gift of salvation offered by Jesus Christ, along with the need to evangelize communities and people which/who have not accepted Christ so that they might also be saved from their sins. Being moral on earth, while still important, is secondary to the question of eternal souls.
If Christians must demand this quality from their leaders in order to not be hypocrites (according to you), then it follows that they must reject the leadership of ANYONE who is not a Christian. You cannot take to some wishy-washy re-definition of Christian values. And given that you probably do not identify as being Christian, it's laughable that you think you are qualified to define what true Christian values are.
But to call yourself a Christian, which is all about morality and following the teachings of Christ, and then vote for a candidate who is obviously immoral, is grossly hypocritical to say the least.
You have made your position clear and you will not budge. I have made my position clear and I will not budge. I can't imagine anything fruitful coming out of a shouting match where we autistically rehash the same talking points over and over again.
I’ve also seen this interpreted to mean that Christian’s should follow the law and obey governmental authority.
Again, if Christians must submit to the government without regard for who their leaders are, then it follows that Christians are not obliged to only support Christian or even particularly moral leaders.