I don't think it should be mandatory for citizens. My larger point is about a player doing something to help...or harm...his team.
There are a host of supplements you can buy legally or illegally - it's up to you if you want to take them, but when your job and your team depends on being tested for banned substances - a player has to put the team first and have control over what they put in their body.
If getting vaccinated allows relaxed in person protocols from mini camp, to training camp, to position group meetings - it's in the team's best interest to get their players and staff vaccinated. You don't want to be the team put at a competitive disadvantage because you had to sign a kicker off the street in the middle of the season, or a QB for that matter.