[TASK] Migrate FormEngineLinkBrowserAdapter to TypeScript

A human-readable diff of the applied changes can be inspected on:
https://review.typo3.org/c/Packages/TYPO3.CMS/+/72795/1/Build/Sources/TypeScript/backend/Resources/Public/TypeScript/FormEngineLinkBrowserAdapter.ts

No functional change to keep the diff minimal and prepare behaviour
as-is.

Releases: main
Resolves: #88435
Related: #96425
Related: #96323
Change-Id: I512739d5827e2951ac6a63312de380870bdebd8d
Reviewed-on: https://review.typo3.org/c/Packages/TYPO3.CMS/+/72795


Tested-by: core-ci <typo3@b13.com>
Tested-by: Benni Mack <benni@typo3.org>
Tested-by: Jochen <rothjochen@gmail.com>
Tested-by: Benjamin Franzke <bfr@qbus.de>
Reviewed-by: Benni Mack <benni@typo3.org>
Reviewed-by: Jochen <rothjochen@gmail.com>
Reviewed-by: Benjamin Franzke <bfr@qbus.de>