In my opinion, neither has more influence/clout than the other, but we can all have different opinions on this, likely as a direct consequence of our varied backgrounds and experiences.
However, what I don't think is disputed is that it's clear that both have more influence than they should have in football etc. It’s the stain of Central Scotland’s religious tribalism that gives rise to, feeds, and appeases that influence. It's pervasive, nauseating and embarrassing.
Much like my personal views on church and state, football and religion should be at a stage now where it can be and should be separate, but that's almost impossible for football culture in Central Scotland due to its social and political history.
As a consequence, the collective Catholic and Protestant victim/conspiracy/etc voices will always be heard the loudest in a Scottish football context, and that's unfortunately how it is these days.