We've already been to a few places that meet the criteria.
Buckeye Regional Park in Maricopa County, west of Phoenix, AZ.
We spent the entire month of February, 2017 at Buckeye Hills and really enjoyed every day. We found a terrific pull through site almost exactly one mile down Robbins Road, on the right. The weather was perfect, our closest neighbor was a quarter of a mile away, it was beautiful. The park is patrolled daily by law enforcement and there's trash pickup. The town of Buckeye, with water, dump, grocery stores, Walmart and lots more is about 10 miles away. Cell phone coverage is fast.
We went on bike rides almost every day (as in dirt bike). After the hard packed and oiled roads stop, there are miles and miles of jeep roads. The scenery is so varied and beautiful. Cactus, the Gila River, Buckeye Hills. And the riding is really fun and technically challenging enough to keep it interesting for Lon. We went up and down washes, up hills, through damp flood plains. It was so much fun zipping around on the dirt bike.
At least a couple days a week, I rode by bicycle on and off the paved roads. I never had any safely concerns out walking by myself.
Rockhouse Trail Road near Borrego Springs
I was excited to visit this area based on the excellent campsite review by Wheelingit.com It's the desert and the actual camping area is pretty nondescript. But there are mountains in the distance and hills worth hiking up almost outside our door. Borrego Springs is about 8 miles away with a good, surprisingly well supplied grocery store. Anzo Borrego State Park is all around this area with a variety of hiking. We didn't have our dirt bike when we were here, but it's well known for its bike riding. Water and dump are available at ABSP for a small entrance fee.