WI the POTUS was elected by Congress? This would be implemented in the Constitution. The President has the same powers as OTL, and can't be removed until his 4 year term is complete. Congress can vote for anyone who meets the eligibility requirements, it doesn't have to be a member of Congress.
I'm thinking that it's the House that votes, with each member getting a vote (not voting by states). But it could be done other ways.
Perhaps this happens because the founding fathers are worried more about the mob.
I'd still expect the first years to be dominated by the Federalist vs. Democratic-Republican discussion. However, under this system we could see third parties gain influence. I'd think a western collation would arise fairly shortly.