The US is sending their military payloads with a US company they have invested in, and most likely have special, undisclosed deals with.
It is not just "a commercial launch provider". I don't expect the US to launch their military payloads with Chinese rockets, private or not, in the same way I don't expect China to use SpaceX for their own military payloads. Same thing for the EU, they prefer to send their military payloads with their own rockets, that is Ariane.
Ariane is private too, it is also a commercial launch provider. It is heavily subsidized by the EU member states and not as competitive as SpaceX, but from a national security perspective, the situation is similar.
It is not just "a commercial launch provider". I don't expect the US to launch their military payloads with Chinese rockets, private or not, in the same way I don't expect China to use SpaceX for their own military payloads. Same thing for the EU, they prefer to send their military payloads with their own rockets, that is Ariane.
Ariane is private too, it is also a commercial launch provider. It is heavily subsidized by the EU member states and not as competitive as SpaceX, but from a national security perspective, the situation is similar.