Freenom (the provider of those free domains) is, in fact, a legitimate company. However, they can take your domain from you at anytime, for any reason, such as if your site is getting “too much” traffic. I believe they do (at least somewhat) touch on this in their terms of service. I have used many of their free domains in the past for various projects, and I haven’t really had much issues, however I have heard horror stories about peoples’ domains being cancelled and used for advertising purposes.
Basically, yeah, Freenom has several strings attached for their free domains, and how they make money (cancelling and parking popular domains) is pretty sketchy.
I will agree that a more sustainable domain provider (and possibly web host, because I believe that’s provided by @KatiaSishost, a volunteer, now) would definitely be necessary for further expansion. I’ve personally looked into Namecheap, which seems to have pretty good prices, but I’ve never registered a domain with them.