[TASK] Use AJAX API in Install Tool

This patch updates the Install Tool to use the new AJAX API introduced
with #89740.

Resolves: #89740
Related: #89738
Releases: master
Change-Id: I0e9e3b43a375b0ec0105ce8ce2db128c0aaaf857
Reviewed-on: https://review.typo3.org/c/Packages/TYPO3.CMS/+/62373


Tested-by: Anja Leichsenring <aleichsenring@ab-softlab.de>
Tested-by: TYPO3com <noreply@typo3.com>
Tested-by: Jörg Bösche <typo3@joergboesche.de>
Tested-by: Susanne Moog <look@susi.dev>
Tested-by: Daniel Goerz <daniel.goerz@posteo.de>
Reviewed-by: Anja Leichsenring <aleichsenring@ab-softlab.de>
Reviewed-by: Susanne Moog <look@susi.dev>
Reviewed-by: Daniel Goerz <daniel.goerz@posteo.de>