Yes, the Roosters have won a lot of comps, but we don't ever see them use a phrase like "pride of the league" to describe themselves; it's incredibly self-aggrandising and nobody likes that.
Roosters and Rabbitohs are vastly different organisations. Roosters are a franchise that rebranded to Sydney because nobody in the eastern suburbs really cares about them more than the Swans, and most of their fans don't even live there, so they had to represent a wider geographical area. South Sydney still has a traditional club mentality, people still fly Rabbitohs flags everywhere in places like Alexandria, Marrickville, Maroubra, and the like. It makes sense one fanbase would be more fanatical than the other.