[DOCS] Update linkvalidator codesnippets

* Make the code-snippets auto-generatable
* Remove API for internal class
* Fix references

Attention: Backport need manual changes.

Resolves: #103317
Releases: main, 12.4
Change-Id: I79686223ee01faff9070612988a25c6f2da0e040
Reviewed-on: https://review.typo3.org/c/Packages/TYPO3.CMS/+/83297


Tested-by: core-ci <typo3@b13.com>
Tested-by: Lina Wolf <112@linawolf.de>
Reviewed-by: Lina Wolf <112@linawolf.de>
Tested-by: Chris Müller <typo3@krue.ml>
Reviewed-by: Chris Müller <typo3@krue.ml>