Where to begin?  Lets start with the "infallible teaching".  Only those teachings which are expressed ex cathedra are infallible.  There have been two of them, thus far, and neither discusses the priesthood.
Jesus chose twelve apostles (that is messengers) who represented the Twelve Tribes of Israel.  He neither chose nor ordained any priests.  There were already an abundance of priests in Israel and He showed no particular fondness for them.  Mary's vocation as the Mother of God would have dovetailed nicely with being an apostle (messenger) and we have every reason to believe she helped spread the word after His ascension. (I doubt she spent her remaining years knitting.)
There is no biological, theological, or psychological reason why women can't be the servant-leaders which priesthood requires.  If we are "all one in Christ Jesus", either ordain women or stop baptizing them.