To my understanding, capsaicin is in the flesh around the seeds, but the seeds themselves are not spicy and tend to add sweetness. Is that correct?

I too have read that, but my experience is completely different. Nothing burns like the seeds.

That jibes with my experience. Leaving the seeds in seems to create a hotter dish.

AFAIK, the seeds tens to be coated in the oils from the surrounding flesh and bring heat for that reason, but less than the ribs themselves. I don't think there is heat in the seeds, but that's something of an academic distinction in most cases.