WOTLK Mac client not working?

I just signed up and finished downloading the WOTLK client for Mac.

Upon starting it, I get the following message:

"World of Warcraft" needs to be updated. The developer of this app needs to update it to work with this version of MacOS. Contact the developer for more information.

I'm running macOS Catalina Version 10.15.3

Does WOTLK client not work on this version (it's the latest one)?

32 bit application support was removed in MacOS Catalina. And wotlk client only has 32 bit version. So you can't play it on this version of MacOS. Your options are downgrading MacOS or installing Windows.