“whence comes the idea that women need a “constant reminder?”
From the Seforim HaKedoshim, as mentioned.
“wedding rings for men became popular when wives figured their husbands going off to war needed a “constant reminder” of their status.”
That is antithetical to us Jews, as you said. Chukas Akum for men to wear rings. And the idea of a married man having a “reminder” not to do something wrong is itself not Jewish. Single men are also not allowed to do anything wrong. The fact that a man is married, in Judaism, does not change much the standards of what is appropriate or inappropriate. Married women have a special status, eishes ish, that men don’t have an equivalent status of.