Your cookie approach requires:

> If you are using the very latest version of Safari on either the recently released OS X 10.15 or iOS 13.4, Apple has just changed the behavior of Safari related to third-party cookies, blocking all of them by default. We are implementing a solution, but in the meantime please visit Safari Preferences, Privacy section, and uncheck "Website tracking: Prevent cross-site tracking" to enable the accessibility cookie to function as expected. [0]

[0]: https://www.hcaptcha.com/accessibility

So while you're patting yourself on the back for not "being like Google", your accessibility workaround exposes blind users to third party trackers like Google.