Same issue here. Everything was fine before 16.3 update on Big Sur 11.7.3 now the start page is showing grey icons instead of the .png files.
So far I’ve tried:
Resetting caches (including via the developer menu)
Deleting the .db caches manually
safe mode restart
Still, nothing. Tech support informed me that they are aware of problems with Safari running on various versions of MacOS regarding the favicons/start page icons. Advised to wait for an update. It seems this problem has been occurring for some time so I’m not holding my breath for a fix anytime soon. Frustrating as I use this feature all the time and without the websites logos to quickly see which one I want it makes the feature redundant.
This is a pretty major functionality issue which should be addressed asap. Fingers crossed.