One approach to paying for a group is to set it up as a business run by the group owner. I write books and make a little money at it. When I sold my first book I had my tax accountant set up a writing business for me. He designed books for the business (actually a spreadsheet) in which I record income and expenses for the business. That simple spreadsheet is all there is to it. No LLC or other business entity. So simple he just threw it in it as part of doing our taxes.
Some years, the writing makes a little, most years, it breaks even or loses some. The years it loses, the loss decreases my taxes slightly. The years it profits, I pay more taxes. Also only slightly.
I suspect the same could be done for operating a group. It's simple and a lot less trouble than establishing and maintaining a 501(c)(3) non-profit. Your members could not claim their membership as a charitable donation, but in my experience, charitable donations have to be substantial before they affect your taxes. Unless you plan on making money off your group for a charitable purpose, rather than just paying groups.io fees, I doubt that establishing a 501(c)(3) would be worth the trouble. If Mark ever wants proof that you are not running a group for profit, showing him your books ought to do it. If you do show a profit, more power to you!
This is just my personal experience. I'm not a lawyer or a registered accountant, so don't take my word for it.
Best, Marv
Marvin Waschke