Remove Public API page

[akeller]

Description

Remove the public API page. Note - this doesn't mean the public APIs are removed or not available, but this page is difficult to keep up-to-date and inconsistent with our current APIs (example - versioning is not consistent across APIs, some are unversioned).

Long overdue follow up, prompted again here #2617 (comment).

This was not fully backported in this PR. We should discuss in PR comments whether this needs to be backported to all supported versions (likely yes).

When should this change go live?

• This change is not yet live and should not be merged until {ADD_DATE} (apply hold label or convert to draft PR)?
• There is no urgency with this change.
• This change or page is part of a marketing blog, conference talk, or something else on a schedule.
• This functionality is already available but undocumented.
• This is a bug fix or security concern.

PR Checklist

• I have added changes to the relevant /versioned_docs directory, or they are not for an already released version.
• I have added changes to the main /docs directory (aka /next/), or they are not for future versions.
• My changes require an Engineering review, and I've assigned an engineering manager or tech lead as a reviewer, or my changes do not require an Engineering review.
• My changes require a technical writer review, and I've assigned @christinaausley as a reviewer, or my changes do not require a technical writer review.