At this point my problem with using it as a web app or just having messenger.com open is that there's no indication of new messages, either on the tab or the taskbar. It's really annoying.
It's true that you won't get anything on the taskbar or toast notification with only a browser-based app or browser tab, because browsers don't connect to that Windows native functionality. The old app does give me toast notifications, and if a video/audio call comes in, it will display a popup where I can click to answer it, even if the app was not "open" at the time (it has a memory-resident component that monitors the communications). That's exactly what I want it to do.
But if I leave a tab open in Firefox with messenger.com in it (and it is not the active tab/window), I do get a sound when a new message comes in, and the tab title will usually (curiously, not always) alternate between the name of the conversation that is currently selected and the name of the conversation where the unread message is. Do you not get a sound?
I get a sound, yes, but if I've walked away from the computer and come back, I'm still not in the habit of checking to see if I have messages. The sound doesn't really help me if I'm not around to hear it.