There has been, so far as I can see, no clear explanation of how "that institution (would) be weakened by" any granting of marriage rights to all!

They may SAY that, but it is subjective rhetoric....all it does is express an attitude. If "weakened" is to be used in this context, they would have to show that their marriage was somehow less valid/meaningful/real/...etc. All such prohibition CAN mean is "I don't like it!"