So, to my best understanding, the reason we can't get the "Happiness Ring" seems to be caused by the hidden counter that enables it to begin with. The wiki states that the counter is saved to your game folder, (Which I think means the folder the game is in, I really HOPE it doesn't mean that a floating counter is made in your computer outside of the folder the game is in without your consent), so if you go over the proper amount, you can never get the ring, which seems to mean such on ANY save file. And while I failed to actually remember if the option that is supposed to show up after doing the necessary action 3 times was there from the very start, it seemed to be for other people. To me, this says one thing: whoever uploaded the game folder had already gone into the options menu and selected "No" to deactivating enemy animations at least three times, probably more. If it was only 3, then when starting from the download, then you would need to do the thing EXACTLY 1 time, no more, no less. It could have also been done 4 times (possibly to get the happiness ring for the person who uploaded it originally), or potentially MORE than 4 times, which permanently locks anyone using this sites available downloads out of EVER getting that item.
Note: When I say the original uploader, I don't necessarily mean the person who uploaded the game for this site, for all I know, that person downloaded the game from the net too, so I can't make any suggestions on that front. I don't actually know how many times the files might have basically hot-potatoed before reaching the audience of f95zone.
I could also be wrong about literally all of this, and it might be some kind of issue with the English translation or something (Saw that "suggested" in the comments on that wiki page).
Edit: Just downloaded the game again for a fresh folder, and unless the idea that the game makes a floating counter just SOMEWHERE in your computer is correct (I really hope not), you very much cannot get the happiness ring...well, MAYBE if your OS is windows 7, I saw that on the wiki (in the comments, one person was able to get the thing to work on windows 7, but not 10. That might have been caused by how they got the game, but I have literally no way of knowing).