I think one of the reasons they stay is inertia.
Shh, you’re only supposed to complain about all the old people in the Congress not show you understand how the Congress actually works.
They would have to vote on a new leader. That could be a bit scary.
I can answer the question a little with respect to Diane Feinstein. Feinstein sits on the Senate Judiciary Committee, which is the committee that allows Joe Biden to continue to appoint new judges to counter all the damage done by Trump and McConnell appointments during Trump’s term.
Democrats tried to temporarily replace Feinstein on the Committee while she recovered from illness. Republicans blocked the effort and made it clear they will do whatever is necessary to prevent Biden’s appointments if Feinstein leaves her seat on Judiciary. So basically, she can’t step down without ruining Biden’s ability to continue to appoint judges.
Republicans will also not permit her replacement assume her position on Judiciary.
These things are never as simple as we’d like them to be.
If McConnell does step down/retire/die/get Raptured, can he pick his own successor or would there be a fight?
I don’t know. John Quincy Adams did exactly that in the House. Why not the Senate?
There are currently 3 Republican senators vying to be McConnell’s replacement should it become necessary to replace him: John Thune, John Cornyn and John Barrasso.
My money is on Thune.
I’m not saying dying in the harness in public is wrong, or without precedent, but that IF (big IF) legacy matters, it’s not the best idea. Very quickly the circumstances of your death can overwhelm your actual achievements, however considerable (evil though I judge them to be, MM has plenty).
Of course, that doesn’t apply to scholarly works, but the public? Yeah. Then again, as I implied with my italicized “power” reference earlier, no, I don’t think his posthumous legacy is one of his top 3 motivations.
Mine is on John.
I’ll bet you’ll win. 
Wow. A choice of three toilets.
Poetic justice of a kind.
I couldn’t have said this better myself except for “delightful” being hardly a strong enough word. In my opinion McConnell was the most evil person in American politics before Trump and perhaps even since.
In terms of his Senate seat, Kentucky law requires the Governor to choose his replacement from a list of three individuals prepared by the Kentucky Republican Party.
In regard to his position as Minority Leader, his successor will be whomever can secure the votes of a majority of the Senate Republican Caucus. It’s all done behind closed doors, so we’ll be spared the public circus of the Speaker’s race earlier this year. As @Aspenglow notes, the “three Johns” are most commonly thought to be best positioned to take up the mantle as they’ve all held senior party leadership positions in the Senate. But strange things can happen in a leadership race, and each Republican Senator will try to cut the best deal he or she can in return for their vote.
How quickly is George Wallace forgotten!
Let’s not overlook Newt. 
George screwed ‘those people’ in Alabama, but AFAIK didn’t have a lot of actual affect at the national level.
Newt on the other hand is in the running, but McConnell’s packing of the courts at all levels dwarfs anything Gingrich managed.
I swear, McConnell’s next ‘episode’ is going to be a complete collapse on camera and no one needs that. Not the least of whom, McConnell.
Newt’s small potatoes compared to the damage Mitch has inflicted on this country.
All previously employed by Yoyodyne Propulsion Systems, oddly enough.