Implement doc checkers

[listx]

Reference: https://github.com/kubernetes/test-infra/issues/24821#issuecomment-1057524854

As per the design doc, we should implement automated checkers to make sure we uphold minimum quality standard for both existing documentation as well as newly proposed documentation.

There have been some discussions in the design doc already, but feel free to comment on here about which checker(s) to implement first.