It may very well be broken.
With the code copied and pasted from the docs, and form detection enabled, I’d expect it to work.
I also note that this user appears to be reporting the same issue:
https://answers.netlify.com/t/static-html-form-not-recognized/129499
The last thing I can think to ask, is how are you deploying the site?
Is it linked to repository? deployed via CLI? ‘drag & drop’?