I don't know what you mean. All I do, is, as soon as a game is over, before I click on anything, I click (touch select) the options icon. Then, I select `share pgn`. Then, from that list I select, `copy to clip board`. Then, I go to one of my stand alone chess apps, like scid, or, Analyze This Pro, and `paste from clip board`. This puts the game into the app, then I save the game to my device or sd card. This is working with Android, but pc would be similar.
Why doesn't stockfish import the PGN's from this site?

Some programs have issues with the v3 PGN download, or at least have in the past and I have one known issue. Which GUI are you using? Stockfish is an engine and must run in a GUI.
I use SCID vs PC and have found it works fine for analysis but the FEN string/tag, and maybe the Setup one, cause auto-analysis to skip identify when leaving book when I have that option selected.
You might be seeing something similar
The pop-up pgn's from sample games or reports?
I can't get it to work...