“I’m curious how remaining silent in shul is a zechus for these things any more than keeping any other mitzvah.”
Simple. It’s a zechut for getting your prayers answered. When you talk in shul, you show that the shul is unimportant, prayer is unimportant. Hashem responds accordingly.
There is a source in the Achronim for this, but I can’t remember exactly. Basically, the Acharon asked Hashem why there are the pogroms. The answer, given to him in a dream, was talking in shul. Contemporary rabbis explain that the talking didn’t cause the pogroms, but talking ruins the ability for prayers – like for protection – to he heard.