[TASK] Refactor init() function of ElementBrowser

Splits huge method (~450 lines) into several small methods,
in the similar way it was done for BrowseLinks and SelectImage.

It also removes some duplicated code between these classes.
This patch is not changing functionality, just moving code around.

Resolves: #55983
Releases: 6.2
Change-Id: I916058485acb9d9abe0086785763edc721a360f0
Reviewed-on: https://review.typo3.org/27631
Reviewed-by: Markus Klein
Tested-by: Markus Klein
Reviewed-by: Christian Kuhn
Tested-by: Christian Kuhn