The registrar owns the domain then, not you. I made a root comment about Njalla which offers this service.
edit: Well you could use fake data. Still risk losing claim to it. I tried doing this with name cheap and they figured it out somehow that the info I gave wasn’t real. Was years ago so I don’t remember what I put in. My guess is it was one of those soft credit lookups (where did you live between X and Y?). Could also be misremembering.
The DNS is cheap. Something like a penny a day for privacy. I typically paid like $50 every 2-3 years for both renewal and DNS on a couple domains.
Was kind of a PITA to be honest as I remember their login process was a little weird. I eventually moved on when they were slow to adopt some of the newer TLDs.