PBA: It depends where. I expect a Rabbi in Lakewood to be significantly more learned than his congregation and therefore more learned than other average Rabbis in 100-member Shuls. All else equal, I expect the average Rabbi in random out-of-town places to be equally learned, “modern” or “Yeshivish”. In my experience, that pretty much holds true. There is a higher expectation in more learned places for the Rabbi to be more learned. And I do think that it’s a misconception (especially nowadays where even “modern” guys spend a year or two learning in Israel and continue learning on some level forever) that the average “modern” Baal Habayis is significantly less learned than the average Yeshivish Baal Habayis. It’s certainly true in the more Baal T’shuvah modern communities, but is much less true in average communities.
In short, in my experience, the average young new Rabbi coming out of YU, Ner Yisrael, Chofetz Chaim, and even Lakewood has about the same level of knowledge, both in Halachah L’ma’aseh and in general Shas knowledge. (Actually, the YU and Ner Yisrael Rabbis usually are slightly better on the daily L’ma’aseh stuff but that is counteracted by being slightly behind in general Shas knowledge; but the differences in both, on average, are negligible.)