Can’t help you with goatheads and tumbleweeds.
Unpaved roads generally isn’t much of a problem.
It helps to learn to recognize what caused the flat.
Penetration flats need another approach than pinch flats.
Bicycle tires are very different from car tires WRT pressure retention. Even on MTB widths, topping up weekly maybe sensible/required.
Get a floor pump with a pressure gauge together with your new tires.
Despite that, hitting a bad pothole/curb/rock can still give you a pinch flat.
Knowing how to ”go light” - get off the saddle, knees and elbows bent, let the bike move beneath you - when you pass a bump is a worthwhile skill to learn.
Sitting like a deadweight will be harsh on both bike and butt.
I’ve been quite pleased with Conti Sport Contacts, but they might not be rugged enough for your use.
There are plenty of tires out there with different kinds of (penetration) puncture protection. There are tire liners(Mr Tuffy) that can be used. They’ll all cost you some in rolling resistance.
I’ve used solid tires and didn’t think they were entirely horrible. But I have mostly decent roads.
I haven’t tried the solid inserts due to the way rims and tires work. Tires need part of the bead to be able to slip into the deepest part of the rim to get the last of the bead over the edge.
With inflatable tubes, this is generally not a big deal. With solid inserts, it can be a considerable challenge. I’ve seen some videos. Really squishy ones may not be so bad to mount, but will have high rolling resistance.
Tubeless have considerable recovery ability WRT penetration flats, but setting up, particularly if rims aren’t intended for it, can take some time.