Rav Moshe explicitly says it is assur, as I linked earlier. There isn’t “absolutely no halachic question”. “
Perhaps that is because when one is wearing Tefillin one is not permitted to do anything that is not b’kedusha (which is why men do not have to wear it ALL day, as is the ideal). A woman’s role, which primarily consists of being the aleres habayis and rearer of the new neshamos that she helps bring into the world, must of necessity get her hands dirty. She is touching things that while are of a holy endeavor, are nevertheless still found in dirty diapers, shmutz, and such, and therefore it would not be fitting for her to have a chiyuv to wear Tefillin, as at any time of the day, she could be called up immediately to fulfill that aspect of her role, the tafkid, I might add, for which she was designed by Hashem. The fact that some women are not content with that being the case, and who somehow feel they are being cheated out of the men’s chiyuv, is a sad commentary on what has happened to our society over the last 60 years.
I don’t believe women should be chattel, or belittled, or made to feel that our role in society is somehow less valuable than a man’s, but neither do I want to be a man, and in fact I am very grateful to be doing what I do, instead of dragging myself out of bed at 5 AM no matter what, to get to a minyan every day. For women who do not yet have the responsibilities of home and motherhood (or who disdain those responsibiities), I reiterate what I have always said – let them do the specific mitzvos and chiyuvim that they already have, properly, and then we’ll talk about the ones in which they have NO chiyuv.
I always learned btw, that while there is no chiyuv for a woman to be wearing Tefillin (b’tzniyus), there is no issur, either. She just does not get a s’char mitzvah for it like a man does, because she is not metzuvah.