Yes, I think that could work. Currently, we have a report that displays the notifications as markers, and the hover-over presents the information for that single Notification (I blurred out the details, but it's basically the Status, etc.):
You are correct that this can get messy when there are a lot. Here's a snapshot of a busier period:
That report is intended to give an overall sense of how busy the period is. The more boxes, the busier it is.
Having said all of that, your suggestion may be preferable. I think what you are talking about is a "COUNT" presented on each day. Possibly color-coded? I mocked this up in powerpoint. 
The Blue 15 would represent one count of Notifications, the Red possibly another count. So the datasource could have several Key Figure columns (Number of Outstanding Notifs, Number of High-Priority Notifs, etc.) and then each could be presented on its date.
The only question is, how would the hover-over work? Would it show the list of Notifications? That would work well, I think, but if the number is large may be difficult. Maybe by clicking on the icon, I could present those notifications in a crosstab below, just by passing the filter from the calendar selection. That assumes the Component would pass a Selection, not sure how difficult that is to create.
This is a mockup of the datasource, as I see it in your suggestion:
