[CLEANUP] Make Service_Db::getDatabaseConnection public and use it

Resolves: #64039

Change-Id: Iabca5466ca440f14d83062ed2fbcb57c24c1c04d
Reviewed-on: http://review.typo3.org/35640
Reviewed-by: Oliver Klee <typo3-coding@oliverklee.de>
Tested-by: Oliver Klee <typo3-coding@oliverklee.de>