The man rather belongs to the society. But he now lives with a factory girl. His friends tell him : you can't go on leading this kind of live, avoiding your friends, giving up everything that means a darn to you. What about the firm ? Do they know ? They'd be enchanted with this set up !
Man : Especially as my chief drag with them was my social connections. Well, I've resigned.
So, should I understand that his chiefs appreciated his social connections, or on the contrary, that they thought his social connections were not good enough (and now that he's with a factory girl, this is even worse ?)
Chief in this sense means "main" or "primary," not "boss."
Based on this context, I would say that "drag" means "sway." The big thing that got him the job was his social connections, and now that he's thrown that away, he doesn't have much of a chance there, so he resigned. Still waiting for a BE native to help out.