As a general rule, I hate to advocate "exotics" to just anyone. You may well be the exception to most rules.
If you have the time and energy, an exotic bird can be an enormously rewarding companion. All are of varying degrees of intelligence, most are extremely interactive and a few are downright "cuddly". For a good, intelligent, disciplined owner, something like a macaw, a grey, an amazon, or even a cockatoo might be the answer.
They will probably run less than the monkey, require less specialized care, be easier to find a qualified vet for, and there is more "bird stuff" and info out there than there is "monkey stuff" and info. All in all, I'd call it a superior pet in almost every respect.
Almost every bird keeper I know has birds at home, none of the primate keepers has primates. Although many do have birds!
There are many different animals that people discuss here, Chins, Degus, Flying squirrels, etc. I must admit I have little or no experience with most of these. Everyone hereabouts seems to rave about them. I did have a chin for a few years. It was a sweet animal, but not the ideal pet for me. I didn't find it all that intelligent, compared to my parrots. That may just have been my experience though...
I personally always wanted to get an African Pouched rat or a Fennic Fox. I've heard some interesting things about them.
As for "Helping Hands", I must say I have had no direct experience with them. However, in my personal circle of aquaintances (among them many primate keepers at several zoos) they are not all that highly regarded. There may be a niche for them, but the animals are "modified" in order to make them more "suitable" for their role. They are neutered, and the canines are removed. They still have a limited usable lifespan, and I have heard stories that there are more failures than successes.
I guess it all depends on who you talk to. The Helping Hands folks certainly aren't going to endanger their fund raising by publicizing their failures, and people who are against having monkeys in that role aren't going to acknowledge the successes.
For a sobering look at what monkeys can do, especially Capuchins, check out this page of "Testimonials" by owners:
The whole website: http://www.petmonkey.info/index.htm
has a lot of good, very sobering information for the potential owner.
It also has a page concerning the legal aspects, and it shows Tenn as have a partial state ban. What that means, I don't know. You asked why the US allows them as pets. That's not really a federal issue. They not allowed to be imported as pets. Many states do ban them. In the states that they are legal, there are many county and municipalities that ban them. I'd check very carefully before spending the money on one.
Glad you are willing to dialogue on this issue..