I think you already know the answer to some of your questions. People have no way of knowing if you're struggling and thinking "Boy, I sure wish someone would offer to help" or whether you're hoping no one does. And it's the good side of humans that wants to help!
I've offered to help people in different situations, and sometimes they've almost snapped my head off. Well, excuuuuuse me for trying to be nice!
I have a friend who is blind. She HAS to do some things herself, such as put things away so that she can find them again. However, once she was being "independent" and trying to help me load the car after one of our little jaunts. She kept trying to put heavy stuff on top of my fragile stuff, etc., and I made her go away. We finally had a little talk. She was hurt because she wasn't "doing her share", but I finally said, "You're putting all the fragile stuff in first! You can't pack the car, and you can't drive either!" It was way better when we were honest with each other.
I don't know if my story has ANYTHING to do with yours, but there it is. If people mock you, it's probably because they feel rebuffed or else feel awkward watching you work (or they're idiots). Sounds like you just need to give them the Readers Digest condensed version of what you just told us.
Something like: "Boy, I wish I COULD let you help me, but if I don't put it together exactly the same way every time, it will self-destruct while I'm tooling down the highway." Well, maybe not that.
P.S. If somebody happened to be standing in your home on Tuesday watching you struggle to change your sheets, he/she would probably offer to help. They see you taking your scooter apart.