I would double check your insurance policy. We just found out my wife's Note 9 can get a no charge battery replacement through Verizon's policy. Your policy may have something similar.
As for the suggestion to upgrade, that's not so cut and dry in my opinion. Prices on new phones are stupid expensive, hard to find, and may loose wanted features. Used phones are also going way up in price thanks to the shortages. My wife's phone also has a cracked screen and we considered just getting something new. The only phone that even came close to being something she would accept (accept, not actually wanting) that is also remotely available in reasonable time was the 512 GB Fold z3. That's a $1,900 phone, no Spen unless you buy it separate, and no SD card slot (otherwise a lower storage version would be a better and cheaper option). We have pretty much decided to go through insurance to get the screen and battery replaced on hers.
It's not a bad idea to consider replacing the phone, but I wouldn't do it JUST because of a bad battery. The Note 9 is still a capable device. Sure it's not being updated anymore, but I don't think you're missing anything earth shattering by not getting Android 12. Going back as far as Android 7 Nougat, I can't think of anything massively different and probably wouldn't notice a difference in day to day use if I compared my current phone to my last one that had Nougat. I personally think there is too much emphasis on it being a near necessity to be on the absolute latest version of Android. Plus, getting a new phone only because of a bad battery is rather wasteful. E-waste is becoming more of a concern these days with the "throw away" mentality.
Kinda makes one long for the days where manufacturers were sensible and didn't seal up their phones so users could replace their own batteries and get more life out of them.