I have the exact same issue. It persists ever since updating to Big Sur 11.3. Even 11.5 hasn't fixed it, super annoying. No matter what I try, the notification badge won't show up again. I can barely find anything on that on the internet, so either few people have that issue or most people haven't noticed or simply don't care. I contacted Apple Support and they told me, the notification badge was removed on purpose, which to me, makes no sense at all. Why would Apple silently remove the notification badge for the Mac App Store only.. I guess they just told me that, because they had no idea themselves. If anyone figures how to fix that, I'd be glad to hear that. Other than that, we can just wait.. I guess it'll only be fixed by updating to Monterey, when it comes out. Apple most likely won't bother to fix this in Big Sur, with Monterey around the corner.