[BUGFIX] Do not block UI rendering when checking for broken extensions

This patch replaces the highly discouraged and deprecated `async: false`
approach by promises to un-block the UI rendering of the browser.

Resolves: #84898
Releases: master
Change-Id: I630fef820784c33758e713e6121c575577af91ed
Reviewed-on: https://review.typo3.org/56830


Reviewed-by: Andreas Wolf <andreas.wolf@typo3.org>
Tested-by: Andreas Wolf <andreas.wolf@typo3.org>
Tested-by: TYPO3com <no-reply@typo3.com>
Reviewed-by: Frank Naegler <frank.naegler@typo3.org>
Tested-by: Frank Naegler <frank.naegler@typo3.org>