Hmm, tough question. Decades ago I spent many months travelling the U.S. with my girlfriend of the time. Had no money so stayed to the cheapest State, County, etc. "primitive" campsites. Hit a KOA, etc. every 4 -5 days to get a real shower, do laundry, etc. Often would just fine a fire road, etc. in the late afternoon, get as far into and hidden as possible, and crash. Up early, leave no trace. Had two really scary nights with people moving around us in the dark, but no trouble.
About a decade later I spent long weekends, vacation knocking around SW Texas, New Mexico, Arizona with my now wife. Usually off the beaten path a bit, never had a problem then, but had protection with me all those times. Rules be damned, where I was, no help was coming and it'd be awhile before they found you.
Nowadays I still do when I'm in a remote place, alone or with family, it just seems prudent. Just understand that if it becomes an issue, there are many places where that is very unwelcome and the authorities will not be pleased. Though the more rural and remote, the more accepted it is. Pay your money, take your chances.
The gear you want for this is all over the hiking sites. Whatever the tent, bag and pad, etc. is all the rage will do fine.
If you keep to the cheap primitive sites, likely your best option. It's an established place and you are supposed to be there.
RT