>>14391609
Rings are portals to the Chaos Emeralds, thus why they let you warp to one in the Special Stage (assuming there are any in the Special Stage at the time). Some rings have more portal energy or whatever than others. Giant Rings in Sonic 3 & Knuckles have enough energy to warp you directly to the Special Stage, while the Giant Rings in Sonic 1 require the additional energy of at least 50 regular rings to be able to do it (note that the power of the Chaos Emeralds themselves was weaker in Sonic 1 due to there being only six of them on the island. The Master Emerald may also have been augmenting things in 3&K). Signposts from Sonic 2, 3, and K, seemingly created by Robotnik, seem to try to replicate this power, with varying degrees of success. Note that the rings' direct connection to the emeralds is made rather explicit in games such as Knuckles Chaotix and Sonic 3D Blast.
So in short, by collecting rings, even regular ones, Sonic and his friends are tapping into the power of the Chaos Emeralds, albeit only a tiny fraction. Thus they can take on a tiny bit of the invulnerability of Super Sonic, but only enough for one hit.
Health bars are evidently because Shadow or whoever you're fighting just doesn't happen to have any rings at the time. Many times in these boss fights they also use attacks that use up Chaos Energy and thus ring power. Attacks that you can't use in single player. So even if they collect rings, they're using that power up in their attacks rather than their defense.
The issue then becomes why you don't have an additional health bar if you get hit without rings. Well that's just because it's a fucking video game.
>>14397097
If rings are conduits of Chaos, it's a neat idea that they can cause Eggman's robots to break free of their programming, or rather, allow the animals inside of the robots to override the programming. The idea that Eggman represents Order and the good guys, particularly the amoral speed freak Sonic, represent Chaos, seems quite obvious, even if the games never got very explicit with it. People always saw it as nature vs machinery, but I think it's more that nature represents Chaos and machinery represents Order. Life, uh, finds a way. When Eggman introduces rings, with Chaos energy, to his robots which are actually means to control animals, that introduces enough chaos to defeat his Order.