Grover Cleveland was ONE president and counting him twice for being nonconsecutive about it is a stupid thing to do regardless of it being "the way we do it"
If something sounds stupid and doesn't make sense how we use it, chances are because at some point everyone just shortened something because "everyone knows what I'm talking about".
Like, I'm sure there's already kids that don't know what LOL actually stands for, just that it means something is funny.
Etc and e.g. are already there.
No one does it better than the Brits tho. Cockney rhyming slang is just the most etymological fascinating shit ever. Like half the words in a sentence sound like random gibberish, but that's the entire point of it.
They do it because they don't want anyone to know what they're talking about, unless they already know what they're talking about.
At the end of the day, words just represent what we agree they represent.
But especially in English, that rarely makes logical sense when just looking at present usage.
That's not even accurate either, since most if not all presidents have made changes to Cabinet officials and other key positions after being reelected.
I've long since given up on the powers that be ever listening to sense on this one, but that doesn't mean I don't reserve the right to grumpily complain every time someone perpetuates it in my presence 🤷
Hundreds of millions of people perpetuating the same obvious mistake and even DEFENDING it using spurious arguments is how things such as the filibuster, carceral slavery, and Donald Trump are still inflicted on the world.