Medicare ... requires any patient that comes to see them sign a document ...More specifically, that requirement is placed only on folks who have signed up for Medicare.
Medicare was established to, among other things, protect older folks who sign up for Medicare from unscrupulous providers who charge larger than customary fees. So - Medicare has established rules and regulations that "protect" folks who have signed up for Medicare coverage.
Older folks tend to get increasingly forgetful as they get increasingly older.
Requiring these older folks to sign a new contract every two years is an effective way of reminding them that the non-Medicare provider they are about to employ is not prevented by Medicare from charging them fees that are larger than Medicare (in its attempt to protect the elderly) will allow. The contract reminds them that they could see a Medicare Provider, who would charge them less because the provider is restricted by Medicare rules if he is a Medicare Provider.
Section 40 of this link covers opt out of Medicare stuff, including private contracts.
https://www.cms.gov/Regulations-and-Guidance/Guidance/Manuals/downloads/bp102c15.pdf