OP is correct that if the average credit-card-PP member had similar usage levels to them (50 visits/yr), the business model wouldn't work, no matter if the banks paid a flat fee per member (PP would be hurting) or per visit (the banks would be losing money). But the average is likely much lower. Especially for members with no lounge in their home airport.

I have two PP cards myself (CSR and Citi Prestige), and I'm not sure I've used either in 2018. I was traveling heavily in Q1, but between MSY (no lounge) and DCA (has choices now, but didn't until this summer). Since then, my travel has dropped way down, and my most recent attempt to use PP was thwarted (Minute Suites was completely full). In the couple of years prior, I probably averaged one visit every month or two.