PF, some common misconceptions about shotguns include one you just typed.
"Pointing in the general direction and blasting away"....
At typical HD ranges, the load hits as a semi solid mass. Spread is not a factor, and the shotgun MUST be AIMED to do the job.The big advantage to a shotgun for HD is not spread, but the incredible amount of energy it can dump into a target.
Measure the longest shot possible in your domicile, add a yard for GPs. Then, shoot at a stationary target at that distance. Bet ya the pattern's less than 5" across.
Next, for a complete tyro, the shotgun is the wrong choice. I found when instructing that the only folks moving past minimum competence (Load, unload, fire, make safe) were those folks that shot recreationally. IOW, a shotgun's an excellent close range defensive tool for those that LIKE shooting shotguns, and did so regularly.
A better choice for a female who will not practice as much as anyone should would be a 38 or 357 revolver, starting with light loads and working up to the hotter stuff later. And maybe a shotgun later also. Many women become fine shots, but it takes dedication and practice, practice, practice.
And,whatever you and she decide to do, a few lessons from a competent and qualified instructor are well worth the money and time. Many folks who shoot well have little idea of how to train someone, and oft have bad habits they pass on.
If you do decide to get the shotgun, start with light loads, and go for good form from the very start.
HTH....