There's another way of removing a president from office aside from elections, term limits, death, or impeachment--military coup. Having an un-elected presidency would likely pave the way for a military coup at some point in the country. Whatever happens after said coup will likely discredit the idea of a president-for-life in the US.
The important political races would be for congress. As soon as a party had a candidate in the White house then the important thing was to retain power in congress. The US politically would mimic the USSR or Egypt today with the President being the figurehead of the political apparatus controlling government. A super majority would be what 2/3 of both house and senate?
We would of had our own version of Stalin or worse in charge of the US since there would not be an checks and balances unless the power of the president become nothing more than a figure head then power rests in congress.
So I am not sure what Hamilton wanted a strong president for life or a figure head president for life with people able to pick the new figurehead every generation or so.
"Stalin or worse" is going a bit too far. An elected president-for-life isn't a good thing, but in the US it wouldn't automatically mean Stalin or whatever (even the worst Latin American dictators weren't that bad). What would possibly trigger some paranoid mass murderer to come to power and kill all their rivals and not be removed? Congress can still override the president's veto and pass legislation the president doesn't want. The Supreme Court can still rule in ways the president doesn't want, although since the court could easily be packed with the president's allies, it would probably function differently than the Supreme Court we know. Maybe make it easier to remove a Supreme Court justice and/or term limits, and of course doing something about Marbury v. Madison or similar cases.
Logically, the Speaker of the House of Representatives would be the one to involve into such a role. But then who is Commander-in-Chief of the armed forces and what role, if any, would a presidential cabinet play in operating the Federal government?
The C-in-C could still be the president, like in European countries where the president is a figurehead. Most of the executive branch would likely fall under Congress's authority (like the Bureau of Indian Affairs, created 1824 OTL), with some of the oldest ones like the Treasury Department and State Department eventually being moved to Congressional control. Congress would probably have full control over the appointment of the cabinet, assuming an amendment wasn't passed which required the heads of these departments to be Congressmen.