Why do they need court confirmation of their own internal disciplinary process? It seems to me it would only become a matter for the courts if it is challenged. Someone enlighten me please on why the NFL would have any standing at all to file anything anywhere, without Zeke making the first move?
Arbitration is a private dispute resolution mechanism. As such, an arbitration decision doesn't have the force of law until confirmed by a court. All 50 states and the Federal Arbitration Act provide mechanisms for seeking court confirmation and enforcement of arbitration rulings. The Labor Management Relations Act, which is directly applicable to this arbitration, also provides for court confirmation and enforcement of an arbitration decision.
So that's what gives the NFL standing. But you're right that enforcement of Henderson's arbitration decision would only become an issue if challenged. The NFL's pre-emptive move for confirmation would simply be intended to ensure that proceedings to confirm (or vacate) the arbitration take place in a more friendly jurisdiction. As you know, the NFL prefers New York to Texas, largely because the Brady case set a strong precedent in New York.
It's a tactical move that the law permits.