Is there an RSS reader that provides this experience?
If I'm misunderstanding your question, please provide more details.
I use elfeed on emacs (https://github.com/skeeto/elfeed) with the time descending sort, and it does exactly what you state in the question. The newest feeds float to the top and the older ones sort to the bottom.
If I let too many build up, I simply mark them as read and they disappear from the list. They aren't deleted, and I can search for them at a later date.
In your case above on not wanting to "Mark as read", elfeed has an filter ex. @6-months-ago that when a feed reaches that age, it disappears from the list. A smaller time scale like @2-weeks-ago might be closer to what you are looking for.
Of course, all of this text is irrelevant if you aren't using the emacsOS tools and don't want to mess with setting them up. But since the RSS spec was invented, I've not worked with any of the readers that fit my personal approach as well as elfeed.
Good luck.
Liferea doesn't! At least not that I can tell. It will do this with unread items but then disappear. There's no option that's just "get all of the feed data, throw it in one place, and sort chronologically".
>I simply mark them as read
I don't want to mark them as read! I don't do this with social media posts. The nice thing about this is that, if things build up, I don't have to feel like I've fallen behind.