Alan Thrall (owner of Untamed Strength in Sacramento) has discussed this a few times (most recently in this video[0]), but basically independent gyms don't really see this effect. In fact January can still be a pretty slow month for independent gyms as members over spend during the holidays (and cancel memberships) or don't want to go out in the cold weather.
Personally as a member of an independent gym for a few years now, I have also not seen an increase in activity at the beginning of the year. Most of the members tend to be pretty consistent and I've been now 5 days this new year at different peak times and seen only one new face. Good independent gyms are very expensive to run.