[TASK] Use new templating strategy in ext:indexed_search

Rather straight refactoring of the indexed_search
AdministrationController towards new backend
templating strategy.

Resolves: #96772
Related: #96730
Releases: main
Change-Id: I889c471f2f7902f5dd407d03d28cb4ac30e21fe9
Reviewed-on: https://review.typo3.org/c/Packages/TYPO3.CMS/+/73335


Tested-by: core-ci <typo3@b13.com>
Tested-by: Stefan Bürk <stefan@buerk.tech>
Tested-by: Benni Mack <benni@typo3.org>
Tested-by: Christian Kuhn <lolli@schwarzbu.ch>
Reviewed-by: Stefan Bürk <stefan@buerk.tech>
Reviewed-by: Benni Mack <benni@typo3.org>
Reviewed-by: Christian Kuhn <lolli@schwarzbu.ch>