[TASK] Simplify EXT:lowlevel Full Search Controller

* Avoids static methods (which were copied from BackendUtility)
* Uses proper UriBuilder everywhere
* Moves some HTML rendering into Fluid
* Removes unused methods

Resolves: #101622
Releases: main
Change-Id: I0b1937cbc71d45aaab952fdfba4acc20a222dd37
Reviewed-on: https://review.typo3.org/c/Packages/TYPO3.CMS/+/80464


Reviewed-by: Oliver Klee <typo3-coding@oliverklee.de>
Tested-by: Christian Kuhn <lolli@schwarzbu.ch>
Tested-by: Benni Mack <benni@typo3.org>
Tested-by: Stefan B�rk <stefan@buerk.tech>
Reviewed-by: Stefan B�rk <stefan@buerk.tech>
Reviewed-by: Benni Mack <benni@typo3.org>
Tested-by: core-ci <typo3@b13.com>
Reviewed-by: Christian Kuhn <lolli@schwarzbu.ch>