A "caretaker" is no solution. This isn't preschool. GOP caucus would be better served resigning all their seats than admit they need a caretaker.
Seriously Wufric, regardless of the issues, how could you possibly trust these people with control of the nation? They can't work with themselves much less opposition, much less another nation. A Republican President would not be effective with this current atmosphere of Republican Congress.
In fact, I'd say a President Kasich for example would be far more likely to be impeached by a Republican Congress than a Democratic one.
Imagine what would happen if a Republican President of the United States actually resigned due to a 30-something-sized House GOP "Liberty" caucus effectively holding him and his party hostage.
Talk about theatrics, and, unfortunately, too much of a real possibility.