[TASK] LiveSearch: Extract handling into repository

To make the LiveSearch API usable in different contexts (e.g. the
record list), the common search handling is extracted into a repository
class, which is now used in the `LiveSearchController`.

In a further iteration, pagination will be added to the repository
class.

Resolves: #100208
Releases: main
Change-Id: I281a7bfa4a99d246bc429ce7a7225c811642e131
Reviewed-on: https://review.typo3.org/c/Packages/TYPO3.CMS/+/78155


Tested-by: Andreas Fernandez <a.fernandez@scripting-base.de>
Tested-by: core-ci <typo3@b13.com>
Reviewed-by: Georg Ringer <georg.ringer@gmail.com>
Reviewed-by: Benni Mack <benni@typo3.org>
Tested-by: Benni Mack <benni@typo3.org>
Reviewed-by: Andreas Fernandez <a.fernandez@scripting-base.de>