I had this idea before.
Maybe could go even simpler, and somewhat like how variations are sometimes referenced in books, and just use the indexes of the choices in the tree without the move numbers. So you have /<game-id>/<move-number>/3.1.1.2.3.1 which means at the first split in the tree take then 3rd branch, at next split (don’t need to say which move number that is, if there is no split in tree keep going) take 1st branch (ie keep on mainline), at next split take 1st, at next split take 2nd. That’s a pretty minimal encoding, and somewhat human-readable. BUT if someone later adds more variations, the meaning of that url can change! How important is that not happening?
Whatever way, you definitely want some UI-clicky not url-typey way to generate. How about on the “Link to Game” dialog from the sidebar if you are currently on a variation it gets a new entry “Current Variation” with the fancy url.