The claim is made that the cap doesn't limit us, but we are limited more than every other team, that is the why it matters who is last. I've pointed out in another thread that we are 25th in actual spending despite having the least cap space available, that illustrates the limitation more than anything else.
Also, most of the league isn't doing that hypothetical thing because they don't need to, and they value the flexibility that not being loaded down with dead money affords a team. The strategy is what it is, but without a doubt the cap creates constraints on teams. We've let enough players leave for mid-tier contracts to see them flourish on other teams while we've signed cheaper "upgrades" that everyone wants to show the door.