If you don’t want to host your own data, take a look at TarSnap
I’m having the same problem. There are bugs about it open with the developers, so presumably it’ll be fixed at some point.
Search Icons are missing again #2227
Icons are missing from custom search engines added from settings #2261
In the meantime, I was able to make my search bar presentable by installing the “Search Engines Helper” plugin and manually adding the search engines that are having favicon problems, then going into the search engine settings and removing the checkmarks for the broken ones.
Molly White also wrote about this in the context of open access on the web and people being concerned about how their works are being used.
“Wait, not like that”: Free and open access in the age of generative AI