I know that in black and white halacha, wearing a fall (with a hat/headband covering the front of the hair) is permitted. Is there any sensitivity that would make it questionable? (Maybe that it’s harder to tell that it’s not your hair, or that the style is less dignified…?)
Yes, as you write, there are Jewish sensitivities that are being compromised.
There are a number reasons why specifically married woman cover their hair, and single girls don’t. One reason a married woman should be extra modest is because she is special for her husband. It is for this reason that she should be further removed from other men seeing her. Secondly, since it is a bigger avieroh for her to sin therefore there is a need for more barriers.
The general idea though is that a married woman because she is married and has a husband, should have a higher level of tznius, and covers her hair in order to make her less attractive to other men. (It goes without saying that a married woman should channel all of her attractiveness for her husband, as her beauty was made for her to attract him, and not others.) Although wearing a shaitel that technically covers her hair and therefore it isn’t seen, is halachically permitted, however the neshoma and spirit behind the halacha is unfortunately lost, when a woman wears a shaitel which makes her more attractive than before she got married, and looks like it is really her hair.
Additionally, many women wear the fall in a way that they expose a finger or two of tier real hair in the front, in order to give the fall a look as if it is really her hair, since what is in front and in back of the headband looks exactly the same. This is a definite halachic problem. There is a great misconception regarding a teshuva that R’ Moshe Feinstein zt”l wrote, which it appears as if he permitted a woman to expose two fingers worth of her hair. This is not true. R’ Moshe never said that it is permitted, rather he wrote that a woman that does it cannot be forced to get divorced, but not that it is a permitted thing to do. In fact R’ Mattisyahu Solomon shlit”a put out a letter in the name of R’ Dovid Feinstein zt”l stating this. That his father zt”l never permitted a married woman to uncover any of the hair that has to be covered. Here is a copy of the letter, https://dinonline.org/wp-admin/upload.php?item=61292 which is brought in Levusha Shel Torah pg. 269.
Livusha Shel Torah 21 3,5,7, Divrei Yoel O:CH 10-11,, Mishnas Yosef 4 O:CH 2.