Yes, you possibly need surgery to shorten the greater horn of your hyoid bone. It’s good the styloids don’t seem to be causing any symptoms. The other things you just mentioned - clicking, rubbing sound of hyoid are more symptoms of Hyoid Bone Syndrome or Clicking Larynx Syndrome than ES.
Yes, resection of your greater horn is a major surgery like the surgery for ES is. It shouldn’t change your voice or give you a big scar if you see a surgeon who is experienced w/ the surgery.
I’m sorry I can’t tell if your C1 vertebra is causing any problems. It has the bony outgrowths I mentioned, but they may not be contributing to your symptoms. I’d say don’t worry about them now. Get your hyoid bone taken care of first. It will take a number of months after the surgery before the symptoms it is causing go away, but that is normal. Healing is gradual. @a_catindisguise is one of our members who had hyoid bone surgery, & she wrote a lot about her recovery on our forum. You could read what she wrote so you know what to expect, but keep in mind we each heal differently & at different rates, so you may have a different post op story than she did. Her diagnosis was clicking larynx syndrome, but that is somewhat the same as Hyoid Bone Syndrome.