The NFL has have-nots, just like college football does.
Yes, but for a completely different reason.
The NFL is set up to be the ultimate parity machine. The draft, CBA, franchise tag and all the free agency rules mean that every team has access to talent. The hard salary cap means that no team can just "buy up" all the best free agents.
If a team is perpetually bad, you can place the blame squarely on the team's management. Because the system is designed to give every have-not equal chance as the haves, if they make good decisions in their GM, head coach, and thereby in their draft / talent acquisition strategy.
This is why the Lions can be doormats for a generation and with the right coach and GM, turn things around. Because there is no inherent systemic process making the rich richer in the NFL and keeping the "poor man" down.
But it doesn't happen overnight. So instead of actually making the structural changes at the HC/GM positions to build a team over the span of 4-5 drafts, bad owners see one coach not doing "enough" soon enough, can him, spin the wheel o' available coaches, watch them not do "enough" soon enough, rinse, and repeat.