[TASK] Move DB related code of EXT:indexed_search to the repository

Make the controller a bit more slim by moving
database related code to the repository.

Resolves: #101958
Releases: main
Change-Id: I6a6f52236dcb6190d20e77f4356424e846702476
Reviewed-on: https://review.typo3.org/c/Packages/TYPO3.CMS/+/81091


Reviewed-by: Jörg Bösche <typo3@joergboesche.de>
Reviewed-by: Oliver Klee <typo3-coding@oliverklee.de>
Tested-by: Stefan Bürk <stefan@buerk.tech>
Reviewed-by: Stefan Bürk <stefan@buerk.tech>
Tested-by: Georg Ringer <georg.ringer@gmail.com>
Reviewed-by: Georg Ringer <georg.ringer@gmail.com>
Tested-by: core-ci <typo3@b13.com>