Interesting. I've tried that, just embedding https://rstudio.cloud/projects and when I click on a public project, I get this (this is accessing that url through an iframe in the course website):
Where the "redirecting" pop-up window flickers but never resolves itself, it just gets hung up here. I get the same behavior if I instead put the project's url in the iframe. When I point the iframe on my website to the specific iframe at rstudio cloud that contains the IDE (here), I get the same screen, albeit with the toolbars removed.
So it looks like the embedding is actually happening ok, it's the authentication that's the issue. In general, I've found authentication done through browser cookies is the most seamless with this use of iframes, so this must be doing something different.
If you're wondering what the use case is here, the course website currently has audio/video chat enabled, so students can collaborate on any content that they're both viewing. If they open Rstudio cloud in a separate tab, they lose the video chat. I know it's a strange set up, but it'd be terrific if I were able to get this working.
Thanks for any other pointers you might have,
Andrew