I was doing some independent research on Islamic laws recently and came across what is permitted for muttah (temporary marriage) for men vs. women. Why is it that men are permitted to have a muttah with “People of the Book”/Christian and Jewish women? The laws did specify that they cannot permanently marry them, but why allow a temporary marriage with them when it cannot become a permanent one? Isn’t muttah working towards a permanent marriage. Why is this option then not allowed for ladies? I am curious as to why the law exists and what the purpose for such a temporary union that cannot progress further would be.

Temporary marriage does not need to lead to permanent. They are different. The opinion regarding allowing only temporary marriage with Ahlul Kitab and not permanent is by some marajie, such as Ayatollah Sistani. It is based on the understanding of the Hadith literature that distinguishes the temporary from the permanent.
Re females, it is unanimously agreed by Muslim scholars according to the Quran that they cannot Mary a non Muslim. We may think it’s because the male may influence his wife but Allah knows best