photoshop elements 2021 version 19 won't work after upgrading to macOS Tahoe

Yes, that is correct: once you update to tahoe:   You can forget about ever using pse 2021 ever again on that computer.   You only option is to roll back to an earlier OS- which I don't think anyone has ever done.

Adobe won't "update" or "fix" pse 2021 to work in Tahoe.
No, you can't get a free upgrade. 

As a previous user you can get PSE 2026 for $79- it's got some really good stuff in it that is waaaaay newer that what is found in your current version. 

Check out Compatibility Issue After macOS Update with PSE 2021
Intentionally* or accidentally**, PSE 2021 will not run under Tahoe 26.

* doesn't pay Adobe to support older versions, or incompatible for technical reasons; license is for 3 years.

** version 26 of MacOS is detected as outside the range ("macOS 10.14 through 10.15 or macOS 11").

[info thanks to jane-e, jeff arola, and Glenn 8675309]