Switch to the standard Lychee link check

[trexfeathers]

Based on this template, which is proven in Iris:

https://github.com/SciTools/.github/blob/main/templates/github/workflows/ci-linkchecks.yml

If you would like a demonstration, here is an action on my fork. The only failed link is the one for this PR, since it did not exist yet at the time of running.

[scitools-ci]

Templating

This PR includes changes that may be worth sharing via templating. For each file listed below, please either:

• Action the suggestion via a pull request editing/adding the relevant file in the SciTools/.github templates/ directory. ¹
• Raise an issue against the SciTools/.github repo for the above action if you really don't have 10mins spare right now. Include an assignee, to avoid it being forgotten.
• Dismiss the suggestion if the changes are not suitable for templating.

You will need to dismiss this review before this PR can be merged. Recommend the reviewer does this as their final action before merging, as this text will continually update as commits come in.

Template candidates

The following changed files are not currently templated, but their parent directories suggest they may be good candidates for a new template to be created:

• .lycheeignore

Footnotes

1. Include this text in the PR body to avoid any notifications about applying the template changes back to the source repo!
   @scitools-templating: please no update notification on: iris-grib ↩

[codecov-commenter]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 90.05%. Comparing base (537eaba) to head (86c5de3).

Additional details and impacted files

@@           Coverage Diff           @@
##             main     #795   +/-   ##
=======================================
  Coverage   90.05%   90.05%           
=======================================
  Files          10       10           
  Lines        2564     2564           
  Branches      420      420           
=======================================
  Hits         2309     2309           
  Misses        156      156           
  Partials       99       99           

☔ View full report in Codecov by Harness.
📢 Have feedback on the report? Share it here.

🚀 New features to boost your workflow:

• ❄️ Test Analytics: Detect flaky tests, report on failures, and find test suite problems.

[trexfeathers]

...
This PR includes changes that may be worth sharing via templating.
...

SciTools/.github#265

[scitools-ci]

Templating

This PR includes changes that may be worth sharing via templating. For each file listed below, please either:

• Action the suggestion via a pull request editing/adding the relevant file in the SciTools/.github templates/ directory. ¹
• Raise an issue against the SciTools/.github repo for the above action if you really don't have 10mins spare right now. Include an assignee, to avoid it being forgotten.
• Dismiss the suggestion if the changes are not suitable for templating.

You will need to dismiss this review before this PR can be merged. Recommend the reviewer does this as their final action before merging, as this text will continually update as commits come in.

Template candidates

The following changed files are not currently templated, but their parent directories suggest they may be good candidates for a new template to be created:

• .lycheeignore

Footnotes

1. Include this text in the PR body to avoid any notifications about applying the template changes back to the source repo!
   @scitools-templating: please no update notification on: iris-grib ↩

I think it's not useful to template .lycheeignore,
as the content is too too specific to each individual project

[pp-mo]

All the changes look great.
Re-checked against the template ✅

But I also did a global search + found that there is still code for the Sphinx link-checking in
docs/Makefile and docs/make.bat.
So is it worth removing those ?