Why not?
It's possible to find someone who would write a prescription for it anyway, as with many off label prescriptions with low perceived risk of harm, but insurance is unlikely to cover it.
Many/most doctors won't do that, though, especially without at least some kind of specific reason (like having recurrent cases already).
But there's a whole "evidence based medicine" thing that many of us usually try to champion, and it turns out the collecting such evidence is something expensive and priority-constrained. Due to lack of suitably targeted studies, there's just not formal evidence that the vaccine will be effective or lasting for us, or on what risks might apply to younger people with more robust immune systems.
There's also just a consideration about rationing the drug itself (it's seen shortages), and so prioritizing availability to more at-risk populations is not totally unreasonable.
All that said, it's not like it's impossible or illegal to get. You just need to find a prescriber who'll sign off for it (they exist), and you might need to pay cash rate for it at the pharmacy.
Prioritizing drugs in the event of shortages is totally understandable, though pretty easily fixable if you give the manufacturers time to plan for general availability (hopefully they would see it as an opportunity to make money by broadening market share, not an opportunity to make money by restricting supply).
We don't know the side effects or the beneficial effects since it hasn't been studied/approved for that (we likely have studied it, but not with enough rigor for FDA approval). It is, in all likelihood, fine, but when it comes to medication, we typically want a study rather than a guess.
It's not clear from the information on local private clinic web pages whether they'll make a fuss or not but I kind of doubt it. It's not uncommon that people manage to pay themselves to prescriptions for methylphenidate and amphetamine, it would be weird if shingles vaccine was more gatekept than those.
Early on there were also arguments that supply was limited, so the age minimum was to restrict supply to those who could benefit the most (frail elderly). But that ship has long sailed.