You'd need a wide screen to make more than 3 articles pleasant, I think. The more you open, the more you'll want to rearrange or be able to increase/decrease or tile vertically.
For example, on gwern.net, we have a very nice Wikipedia popup integration (which is in some ways better than OP - eg we follow redirects and handle dark-mode natively), and while most readers never notice it and will use it in the basic recursive popup mode like https://gwern.net/doc/design/2021-04-01-gwern-gwernnet-annot... , it doesn't do just recursive popups.
It's actually basically a full-blown tiling WM with keyboard shortcuts! You can have an arbitrary number, drag & resize, resize them to fullscreen or aligned to an axis, etc. So you can popup as many as you want to fit on your screen and rearrange them like in this demo screenshot: https://gwern.net/doc/cs/js/2023-09-14-gwern-gwernnet-popups...
For example, on gwern.net, we have a very nice Wikipedia popup integration (which is in some ways better than OP - eg we follow redirects and handle dark-mode natively), and while most readers never notice it and will use it in the basic recursive popup mode like https://gwern.net/doc/design/2021-04-01-gwern-gwernnet-annot... , it doesn't do just recursive popups.
It's actually basically a full-blown tiling WM with keyboard shortcuts! You can have an arbitrary number, drag & resize, resize them to fullscreen or aligned to an axis, etc. So you can popup as many as you want to fit on your screen and rearrange them like in this demo screenshot: https://gwern.net/doc/cs/js/2023-09-14-gwern-gwernnet-popups...