Cap is real. The Saints have lost a lot of players in the last few years and they haven't really been able to be competitive in a while. The Rams are now playing with 3 studs and 50 nobodies (who turned out to be good thanks to McVay). There are teams that navigate it better in the short term but in the long term it's more than real.
If a team can draft really well, it masks some of the cap problems for a while.
The Saints have a few guys they have had for 800 years and just signed them to absurd contracts that they push down the road every year. They have to do that at this point, they can't create cap space any other way. Cam Jordan, Kamara, Ramczyk, Peat, Lattimore, all lifers there, 7 + years. Eventually these guys and others won't be there anymore and will fall on the dead money pile. Admittedly, they've been at this game for a long time, but threading the needle gets harder every year. Every year they either have to have a good draft with multiple day 1 starters, rely an abnormal amount on old players, or just suck at a position. Drew Brees also covered their derriere a lot. They've been mediocre at best for 3 years now.
That being said, there are plenty of other teams that manage the cap well and suck just as bad or worse, so that method isn't much better either.