just upgrade your iMac's GPU.
just a little while back an bootscreen EFI mod is also released, and works great
I've so far upgraded two iMac's and applied the EFI mod to hem, and Mojave/Catalina works fine now..
Check out this thread :
iMac 2009 to 2011 MXM3 graphics card upgrade (Lastest update 10/13/2023 - this is the official documentation page - you found it finally!) This is the single and only documentation post. During the last years we constantly updated this post to reflect the lastet development of vBIOS and OCLP...
forums.macrumors.com
I still hope one day, HD3000 kexts are put back into the Mojave/Catalina patchers to get sleep mode working again, and I think also Airplay mirroring require them for these 2011 iMac's..
Also by default disabling legacy graphics patch on the Mojave patcher from dosdude1, like it is now with Catalina patcher would be great if made possible !