First, technically speaking it is not a violation of 1st Amendment rights. The amendment prohibits government action restricting speech. As a private organization, the NFL can set conditions, and in this case the punishment falls on the member clubs, not the players, which probably further protects the league.
That said, it is a stupid decision,
As far as the bloviations by the President and Vice-President, I would like to refer them to some words from a still in effect Supreme Court decision in 1943, written by Robert Jackson, who was later while on leave from SCOTUS the chief prosecutor at the war crimes trials in Nuremberg.
Those who begin coercive elimination of dissent soon find themselves exterminating dissenters. Compulsory unification of opinion achieves only the unanimity of the graveyard.
The case involved punishing a student who refused to say the Pledge of Allegiance because of his religious faith (Jehovah’s Witness). The case actually reversed, 6-3, an opinion (8-1) from 3 years earlier which had allowed Witnesses to be expelled for refusing to say the Pledge.
The principal by which Jackson ruled, illustrated in part by the quote above, is equally applicable to the anthem as it was to the Pledge.
Let me also offer the most famous words from that opinion:
If there is any fixed star in our constitutional constellation, it is that no official, high or petty, can prescribe what shall be orthodox in politics, nationalism, religion, or other matters of opinion, or force citizens to confess by word or act their faith therein. If there are any circumstances which permit an exception, they do not now occur to us.
Perhaps this country would be better served if those at the top of the administration had some understanding of why we have a Bill of Rights, and at least made a pretense of abiding by the principles contained therein.