For me, it’s not just because of the delay. Quite often I will take a screenshot on my iPhone but want to post it using a Mac, but using this media picker from within Safari I usually find that iCloud photos has not yet updated, and it will not update if I keep this media poker open - I would need to cancel, then choose the upload again. So, I open the Photos app, wait to see that the Photo is there, then export it and just drag-and-drop into Safari, or choose from the exported location in the file chooser view.
Another case is when I want to choose an older photo - it’s just a lot easier to find it in the Photos app.
Lastly, in Big Sur at least anyway, the picker at the top that lets you select a particular album is minuscule and it’s hard to discern that it’s even there. I know now that it is, but the first time I used it, it took me a while to find out where it was (or, really, figure out that MacOS was just not showing it for some reason, or it had been inexplicably removed in Big Sur.)
(Regarding my previous comment about not knowing if this is still an issue in Big Sur, I was traveling without a Mac and just tried for the first time this morning.)
You can all see this behavior by starting a reply here and tapping the “upload” toolbar item in the reply window and ten scroll down to the sidebar to find “Photos”. You can see why it was characterized as being Finder behavior - this file picker sure looks almost exactly like the Finder.