Pensacola weather is, actually, generally good.
The worst part of the year for weather is January/February, and that mostly only limits North Field. South Field helo ops can go on in almost anything depending on the phase of training. If they anticipate bad weather, squadrons can det out (Roswell or Las Cruces, NM are common locations) for a couple weeks.
There's also a much bigger footprint to flight training than you may think. Flight school requires multiple outlying fields (NOLFs) to accommodate pattern work as well as military operating areas (MOAs) that can restrict civilian operations. To find another area that neatly fits these requirements is tough.
Much of the SW is already cut up into MOAs and restricted areas supporting operations out of Pendleton, El Centro, Luke AFB, Holloman, Alamogordo, Davis-Monthan, and probably a dozen other bases I'm forgetting. While it may be possible to squeeze flight school ops in there for a couple weeks due to a det or DFT, the amount of flights that support all of primary on a daily basis is prohibitive. There's just nowhere for them to go.
There's also the issue of what's nearby: Pensacola is a pretty Navy friendly area, but there's often local resistance to new military flight ops moving in. We're loud at almost all hours of the day and (accurately or not) there's concerns about crashes and damage to the local area.
Students also benefit from active ATC airspace. The area around Pensacola/Mobile/Tallahassee is actually pretty lively, much more so than the middle of the desert.
So, in short, though inertia is probably 90% of the reason there are valid reasons for not moving flight training. Pensacola (and Corpus, Kingsville, and Meridian) is "good enough" and the cost and difficulty associated with moving just doesn't add up to making sense.