The NFL isn't really a medical source for where you should be getting information. They have a vested interested, one that's several million per player, to get them back on the field quicker. Also, US. Most people here don't care anyway at this point. Now, to what you said.
First, back in may, this was published. Asymptomatic cases account for 50%, at least. On that front, its not seeming any less infections. Just harder to track and trace.
Second was this posted on the 5th of December of this year. Now, that study isn't for sure on it at all. Its a working theory. The theory being that they are sick for less time, and thus, That is what would lead to them being "less likely to transmit it". The pure fact that they may not shed the virus for as long. So, to that point. Its not that they are less likely to actually spread it. Its that they Might be able to spread it for less time then if they were not infected.
That's the only information I can find. Yes, they spread it. At least half the cases were from asymptomatic. The only thing is that if they are fully vaxxed, they end up shedding the virus for a lesser period of time, but can still spread it just as easily while its shedding.
This is the kind of misinformation that is helping the pandemic stay active. People need to research into things they hear and be informed. Too many are thinking the vaccine is a cure all. Its not. It was never planned to be the Only thing we do until covid had been wiped out fully. They always intended for us to still social distance and mask up. Now though, the vaccine is providing less protections. All thanks to the idiots who let it mutate by not getting vaccinated when they were able to stop it.