We have done it numerous times in Canada. We land at the airport, pickup the RV, head to the nearest hyper market (the Costco on steroids that was invented in Canada decades ago) and we are on the road with the RV and spend the first night after leaving home in the campground.
We take our photo and fishing gear and only the clothes we will need and this is not a problem even with the stupid TSA restrictions. We do make sure to take anything of value on the plane so it will not be stolen by the TSA or airline employees.
The quality of service provided does vary and we have found the smaller family run operations to be the best. They have had the same range of RV choices as the chain operations.
For getting around we go with either a 1-ton 4WD pickup with a slide in camper or a Class B motorhome. No problems with getting into any campsite and for places like Zion and Canyonlands we can park in the regular parking lots at trail heads which would not be possible if we were towing another vehicle. There are B+ motorhomes with double slide outs and these provide a lot more room inside and we don't need the exterior storage space for a 2 week trip.
We are used to chartering boats for a week or two and taking only a duffel bag to the boat for our personal gear so transitioning to doing the same with an RV was no big deal.
What is great is that at the end of our trip we arrive before noon at the RV rental place, have it inspected, and our flying home the same day. We do not waste a week driving to an area and back home again but spend all but 2 days actually camping and exploring.
There are many places to rent an RV in Arizona and I would also check out Las Vegas as well.