A friend and I started a classic car club in the early 90’s. The issue with it becoming a formal club is liability. I can’t speak for bike clubs, but to protect us, we had to have club liability insurance. The more formal you become the more you need to protect yourself.
Running a club can be fun and a lot of work. Club events, club shirts (jerseys), a newsletter. Club dues and accounting. We piggybacked on a larger association. We never had to file anything with the IRS, but we ran Peachtree and provided reports to the members
This may be more involved than a bike club, but once you charge dues, you have to show where the money went.
John