I can't explain why, once you're on the Messages page, doing a search for "Subscription" doesn't whip you right over to the "Subscription" link that's in the control bar on the left hand side of that page (along with all the other control links that are accessible to you as a user, but weren't pertinent). If you have more than one group using the "Your Groups" control to transition between them allows you to stay on the same page you're on no matter what group you're looking at. I tested those steps out, using NVDA on all three of the major web browsers: IE11, Chrome, & Firefox, and I cannot get them to fail. They work each and every time and without being there to see what someone else is doing I can't fathom how they're failing for them if they're followed to the letter.
That URL list you give is not rock-solid consistent and how I wish it was. The vast majority of Groups.io groups match that formula but it depends on precisely when they were created. Groups.io was in very late beta stage when it "hit the air" and things did change. It's worth trying that route, though, and seeing if you land where you expect to.
In any event, my main point is making people aware that there are some very specific controls over messages that can only be tweaked by the web interface. Trying to do some of the "If this, then do you want that sub-option or that other sub-option," is just not easily feasible via e-mail.
Brian - Windows 10 Home, 64-Bit, Version 1703, Build 15063.413
Like all weak men he laid an exaggerated stress on not changing one's mind.