[CI] Automation: have link-check workflow commit the updated refcache

[chalin]

• When a PR's check-links workflow fails because the refcache hasn't been updated, wouldn't be great if the job could just submit the updated refcache? It's already done all the work, might as well save it, no? Otherwise, we typically just end up issuing a fix:refcache command to achieve the same result.

  WDYT @svrnm @trask @open-telemetry/docs-maintainers?

• While building the website for the purpose of checking links and building a new refcache, we might as well consider running some of the other fix commands that are quick to run, such as fix:dict, etc. Also see, [CI] Ensure that fix:dict etc get run as part of checks #6040.

[chalin]

Of course, there's the possibility that we don't want the updated refcache to be committed. In those cases, which are more rare, we can always delete the commit.

[tiffany76]

Giant +1!