A performer on a military base is constitutionally protected from punishment for kneeling during the national anthem, because that base is public and they're acting as a citizen. A member of the military may have other requirements, I'm not really sure.

A professional (as opposed to high school or college, where this argument doesn't work) football player is playing on private property in the employ of a private citizen. That citizen could (and does) have the right to fine their employee for kneeling during the anthem. This isn't true for HS or college players who, by and large, are not acting under a private citizen owner, but are normally playing under the purview of a publicly funded school.

To put it succinctly:

>And they are taking these actions on private property.

Yes, and that private property is someone else's (in this case regulated by the NFL) who can set rules as they see fit.