This is a fantastic question!! I think the small piece that you are missing is that when Connect renders content each day, for instance, it stores the "whole output bundle," as its own object each day (including all associated output files). This content can then be browsed interactively (Navigate to "History" and then an older run time):

The URLs for each of these items look like: https://myconnect.com/my-vanity/_rev157059/
The "revision" part of the URL points to a specific output bundle (and all associated output files).
As a result, day one may be at:
https://myconnect.com/my-vanity/_rev157059/dayone.html
Where day two is at a different _rev URL:
https://myconnect.com/my-vanity/_rev158952/daytwo.html
As a result of this, you don't actually have to name files uniquely in order for this to work 
Further, while rendering the report, you have access to the revision information in your code through environment variables (described here):
https://docs.rstudio.com/connect/user/rmarkdown/#r-markdown-including-urls
Unfortunately, today there is not a great way to list these output bundles for programmatic perusal as you may be shooting to do. Is it possible for you to articulate what you're shooting for here? What do you plan to use these HTML documents for over time, and how will they be referenced?
If you want to decouple the output HTML from the Rmd that produced it, then you could also deploy the static HTML as its own static content item on Connect (perhaps using the experimental connectapi package to marshal programmatic deployment, or calling the API directly).
EDIT: Formal docs on this topic are here: https://docs.rstudio.com/connect/user/report-history/