Just to clarify a few points here.

There is Fast Track security at C but it was put in at the behest of easyJet as part of their product marketing. It's just one channel on the side of one of the two main centralised security blocks. I would be extremely surprised if this was a significant factor for the OP since unlike some other German airports the main C security is pretty fast, and the Fast Track can easily be slower. Either way I bet the OP was through security much quicker than via the A terminal process. As a tip, you can use either block, you can walk through between the two C departure halls.

LCY tries to leave from A, but the problem there relates to airbridges, staircases and passport alignment, and so the options are limited, so occasional use of C does happen. TXL being an old airport means no-one wants to spend money to improve A at this stage of the game.

The lounge issue, well the OP presumably knew about it being in A and just didn't check the timings. It is a five minute hike along the corridor but there seems to have been time, just about, and the lounge agent would have in any case advised when to leave. The OP may have found it better to have another 10-20 minutes in hand though. For this sort of thing, it's always good to check via Flyertalk beforehand, rather than afterwards, or indeed to read up the relevant threads in this forum on the arrangements at TXL.