Guess they haven’t seen this old version with Betty Garrett and Red Skelton: https://www.youtube.com/watch?v=xpDLpz88V-I
In the film Neptune's Daughter the song is first performed by Ricardo Montalbán and Esther Williams, then by Betty Garrett and Red Skelton but with a comic parody twist: this time the man wants to leave and the woman is the host and wants him to stay.
That version with the woman wanting the man to stay the night has been done recently.
Lady Gaga & Joseph Gordon Levitt - Baby, it's cold outside
I guess the difference is that back in the 50’s a man trying to convince a woman to stay the night with him, even plying her with liquor, was seen as “romantic” and normal while a woman trying to convince a man to stay the night with her using the very same lyrics was seen as comical and abnormal in a way, but today in the song’s original lyrics and methods of seduction, the man is seen as a sexual predator but if performed by a woman, the woman is seen as empowered and sexually liberated.
In either case I’ve always seen the song lyrics as, whether the man or the woman is the “aggressor” or it is the man or woman being convinced into staying the night, it is someone trying to come up with excuses not to stay, but not really needing all that much convincing. In other words, they really do want to stay, and not because it’s cold outside.