Average Reds said:
That's a pretty huge strawman you are building.
What people have pointed out is that non-public injury information is extremely valuable to gamblers. Why would the NFL want to create a system that encourages gamblers to seek out that information from teams?
You can believe whatever you want, but the requirement to report injuries predates fantasy football.
To add some specificity to your points: the injury report dates all the way from 1947, when then-Commissioner Bert Bell made it mandatory. Given that Bell spent so much time as commissioner fighting match-fixing and gambling in the NFL, and given that the requirement of a mandatory injury report came in the wake of the 1946 championship game fixing scandal, it's not hard to see how this was part of an anti-gambling measure, which explains the strong "integrity of the game" language around the injury report section of the rulebook.