[TASK] Indexed Search: Remove `searchData` property

The `searchData` property is now removed to get rid of some state in the
`SearchController` class.

Resolves: #102936
Releases: main
Change-Id: I9b49de7b75ad85d317d04dcf27605e5c1a238de1
Reviewed-on: https://review.typo3.org/c/Packages/TYPO3.CMS/+/82652


Reviewed-by: Oliver Klee <typo3-coding@oliverklee.de>
Tested-by: Andreas Kienast <a.fernandez@scripting-base.de>
Tested-by: Benni Mack <benni@typo3.org>
Tested-by: core-ci <typo3@b13.com>
Reviewed-by: Andreas Kienast <a.fernandez@scripting-base.de>
Reviewed-by: Benni Mack <benni@typo3.org>