given that SpaceX is choosing what price they're charging starlink, there's a reasonable argument that starlink isn't profitable either

Does Starlink pay SpaceX for launches?

Yeah, Starlink is about 3.5x the space launch revenue but still only about 0.5x in terms of profit. Falcon 9 is as optimized as a rocket could be, and absolutely owns the market. Starlink is a mostly rural service with global consumer pricing where average monthly rates in poorer countries drag the average down. Starlink government and commercial business, however, is growing quickly and I expect that soon Starlink will be ahead of launch, in terms of income, probably by the end of this year.