Copy the FEN string on the analysis board and then you can use the position
Stand-alone games/images/puzzles

Seems like the answer is no... Copying the fen is obviously not going to do anything apart from giving me the position on the board... From what I can tell, even to get a standard board to play through requires the old site, and even then I don't see any way to get it to spit out a puzzle.

I can't see one. The closest thing I can see is a download button, which will let me grab a pgn or the url of the analysis board tool (in case anyone else would like to know where they can go to set up a board that they can't show anyone! )
There also appear to be some slight rendering issues with a blank tab on the download window, and a completely blank (tab and contents) and misaligned tab to the right of the information tab on the main board.

Thanks, I guess that must be the problem then because that sounds like the weird blank tab I have:
I can click on the tab, and it has no contents.

Clearing the cache didn't help, but knowing where the problem was, and that it had a solution, I figured out the issue is apparently connected to adblock which I had previously enabled in order to make live chess usable. (Ads were causing extreme lag and stuttering, and extremely questionable links that basically looked like they were going to try to install malware on my machine were popping up right in the middle of the board when my opponent resigned that I REALLY didn't want to accidentally click on.)
So problem solved I guess. I'll just have to keep fiddling with adblock until I decide it's time to give the ads another chance.
Is it possible to create a chess "board" on chess.com outside of a forum comment/post or blog/article or whatever, that I can then link to? I can put moves into the analysis board but then I can't do anything with the result. There doesn't seem to be any way to save it or link to it.
I notice that posting a daily puzzle link on facebook will create a simple embedded board and puzzle, and was wondering if it is possible to, for example, comment with a related puzzle that I could make on chess.com somehow...