I poured my heart and soul into trying to fix my cold start issue, and even had VW UK involved. They diagnosed injectors which were sent away and cleaned, this made no difference even though tests (by the company that cleaned them) showed an improvement. After that I replaced:
Plugs
Fuel Filter
I had the coil packs replaced as part of the recall.
Anyway just keep that key turned until its running – some days mine fires straight up, some days I can feel it splutter and as you say takes a while to catch. My car had 19K on it when I got it, it’s not got 64K on it and it’s never got any worse.
I recently had to replace my PCV valve and this has transformed the IDLE, but hasn’t helped with the cold starts.
As your mechanic says, some of the TFSI engines need a few more turns that others and is attributed to the HPFP needing to build pressure on the rail.
I spent quite a while worrying about this, my honest advice to you is to forget about it and enjoy the car