If Kaine didn't run in IRL 2020, when he'd almost certainly be the frontrunner (even if Biden jumped in, I think Kaine would beat him in a primary), then I don't think he runs in 2024 as the incumbent Vice President.
A lot probably depends on how 2018 and 2020 go for Democrats, whether they can stop the bleeding a bit, and how many seats Democrats can pick up (if any) in 2022, and who the incumbents would be, both in the Senate and in the Governor's mansions.
I'd guess that someone like Tammy Baldwin (if she didn't lose in 2018) would be the nominee, otherwise I'd guess that the nominee would probably be a younger progressive type (maybe Ayanna Pressley if she runs for Markey's seat instead of Joe Kennedy III), or an older progressive who can serve as the "next in line candidate" after Kaine declines to run (almost certainly Bernie Sanders-he might even beat Kaine in a primary if both run) would be the Republican nominee.
Republicans would almost certainly either nominate someone from the their 2018 or 2020 Senate or Gubernatorial class. While Cruz would start out as the perceived next in line (not Pence), he'd be a weaker version of Jeb Bush, since pretty much no one in the party likes him. He bails well before Iowa to focus on his reelection bid (and loses that too, even as Republicans still carry Texas at the Presidential level). The frontrunners would probably be Josh Hawley (still elected to the Senate in 2018), Paul Ryan (still Speaker of the House, the only top candidate not in the 2018 or 2020 class, he occupies the Newt Gingrich 2012 of this field), and probably Ron DeSantis (elected Governor of Florida in 2018) and possibly Nikki Haley (leaves office as Governor of South Carolina in 2018).
Not sure who wins the Republican nomination, but no matter who it is, they are favored.
If Kaine ran in 2020 (as in our real 2020 primary) he absolutely would not have been a major force, and Biden could/would have beaten him.
Kaine was adequate as a VP choice for Clinton but he has no base outside of Virginia. He definitely would have struggled to gain traction if he ran in 2020, especially in as crowded and chaotic a field as the one we had.
