Its simply 100% untrue what you're saying. You can see by year exactly what teams are spending in cash. Its all public record and very accessible.....
https://www.spotrac.com/nfl/cash
Then you can compare that to what each team has in cap space
https://www.spotrac.com/nfl/cap
From the links we can see that the Minnesota Vikings have at total cap limit of $296,902,277, yet they've spent $355,514,344 in total cash.
Here is a player example:
https://www.spotrac.com/nfl/player/_/id/47594/joe-burrow/contract/cap
https://www.spotrac.com/nfl/player/_/id/47594/joe-burrow/contract/cash
Joe Burrow signed an extension in 2023. That year he was a $19M cap hit, yet he received $45M in cash that year...on top of that the Bengals had to write a check for his $146M in guaranteed money.
The entire way that teams push cap hits into the future is by paying out bonuses to players now....basically the more cash you're willing to pay a player today the more cap hit the system lets teams push down the road.
The part of what you're saying that is accurate is that the league does have profit sharing which is largely (if not fully) where they get the cap numbers each year from. The issue is that these don't get distributed to teams until mid to late summer, there is no specific date....or at least that is made public. So early offseason extensions and free agent activities absolutely comes out of either team funds or the owners pocket. Teams raising more cash to support their franchises has been a huge topic the last few years, and part of the reason why we are seeing so many teams being sold. The amount of cash required to run an NFL franchise is no joke.