The horse's mouth is highly sensitive. He can probably feel a piece of string.
Yes, he could feel it, but bits are about more than just feeling a pull. Bits are about providing a subtlety and a clarity of cues that other means of control just don't have. You can use pretty much anything to drag a horse's head around by the reins, but with a bit, you can pick up some of the slack out of a rein and get the desired response where with some other options, the horse wouldn't even feel the difference.
Native Americans didn't use metal bits
Yes, they did, if they happened to have access to one. For those that didn't, the most common form of controlling the horse was a "war bridle" which was really nothing more than a thin strap of cord or sinew tied around the horse's lower jaw and through their mouth on top of their tongue. Used improperly, that has the ability to be incredibly harsh. You can do a google search for "indian horse war bridle" and find old pictures of actual native americans riding with just such a headstall.
Plus, the Native Americans weren't the supremely kind and understanding horsemen that people like the Pony Boy guy and disney movies would like folks to believe. Their training methods were abrupt and sometimes cruel, really not too much different from the cowboys in those days. The reason for that is they needed results as quickly as possible, they didn't have time to play with a horse in a round pen for 60 days before they tried to ride it, they had to get their horses broke quickly and get them suitable for war so that they would be safe to ride into raids and hunting buffalo.
Anyway, back to the OP. Step one is to have her checked by a dentist/vet to make sure her teeth are okay and there's nothing in her mouth that could be causing her pain.
Also, I didn't see anywhere that you described the bit that they used on her at the rescue when you rode her and got along well. Maybe you did and I just skimmed over it. What type of bit was it? If you don't know how to describe it, you can do a websearch for pictures of all kinds of horse bits and you can probably find one that looks similar.
Do you have a picture of her in the bit you're currently using? Though I agree with others that a thin twisted bit is one of the very worst options to try because even if there hadn't been trouble before, one of those used even a little bit wrong can cause some big trouble.