Not likely. Apparently "born of a woman" was standard Jewish language to emphasize humanity. E.g. Job 14:1. Also the context is emphasizing the Jesus was a normal Jew. Putting the Virgin Birth in that context would be unlikely.
Think about that statement and the OP's opinion.
Why would anyone supposedly believing Jesus birth was "normal" or "ordinary" have any need to emphasize His humanity?
And if it was indeed believed to have been an "ordinary" conception/birth then what sense would it make to emphasize only the woman when "born of man" would suffice?
No.
We cannot say it is unlikely. If anything the suggestion of your post is not just less likely but makes no sense. That it is a very awkward/odd phrasing absent the belief is precisely the reason so many Protestant and Catholic theologians give that very verse for proof Saint Paul held that belief. Same reason the atheist latch onto that verse to suggest a "non-witness" to the actual birth actually made up the Virgin birth later and that it was Saint Paul who created the myth of a God-Man.
If those understandings were "unlikely" as you suggest then none of those teachings or the atheist claims about Saint Paul would hold water.
Job 14 says using the Hebrew translated as "born of a woman", which I agree Job is clearly referencing an "ordinary" birth of "all men".
Saint Paul specifically, and obviously intentionally, used the phrase "made of a woman". Common idiom - yeah for "
made of a woman" see all the Biblical cross references to virgin birth, especially OT prophecy of a Virgin Birth to which Saint Paul being a well educated Jew in using similar language to those prophecies could only be seen as alluding to them in choosing his wording in Gal 4 - to that of a virgin birth. See NO cross references with similar language to any "ordinary" birth.
Further ask your self why a Jew of "ordinary" birth to two Jews, would need a statement emphasizing that the result was a Jew (or human)?
Interesting that you brought that up given those familiar with the claims about His Birth in those days and not believing those claims suggested to His Face that He was a bastard child. There would be no basis for making such a charge unless it was true and known by those taunting Him that there was a belief that Joseph was NOT His biological father.
BTW and since I mention it and as some apparently think there is no such support, that would be another indirect reference to the Virgin Birth in the Gospels.