Can I add AppleCare 4 months after purchase?

No. AppleCare+ plans need to be purchased within 60 days of the date of original purchase. There is no documentation at all supporting your "odd exception."

No, the window has closed.

Isn't there some odd exception where if you have a warranty repair in the first year after the window is closed, you can then add AppleCare?

There were a bunch of stories around August 2020. Looks like it may have amounted to nothing.

Apple just doesn't bend their rules. They'd have mayhem if they did. Policies are uniformly enforced.