If you can do neutral grip on rings I don't think you can be that far off. Can you do a full range chin up?
I'm not 100% convinced by band pull ups. I use them with the female rowers I work with but I'm not sure we get bang for our bucks in the limited time available. I know Rob Shaul abandoned them and moved to a jump and a slow negative.
A long time ago we had a thread about getting Shaf to 20 deadhangs and after some of the discussions I played around with bands for about a month, result: sore elbows and chaffed knees.
Dave Van Skike said Pendlay Rows really helped his pull ups.
Couple of tips:
1. Look up at the ceiling
2. Try to imagine pulling your elbows down rather than yourself up.
The rationale for reduced gin intake and the knowledge of the perils of alcoholism and attendant metabolic derangement has almost entirely come from physicians and researchers.