Are the US sending military payloads with non-US rockets?
EU countries, not having a reusable rocket at their disposal, will have to pay through the nose for every launch, discarding the entire expensive rocket in the process.
Of course, that will limit their ability to launch satellites into space: the cost of discarding a rocket is high (let's not even start about fairing dimensions and subsequent limits on payload size). Wrecking a sophisticated machine after each use is uneconomical.
Meanwhile, the US is galloping towards much cheaper launchers. This means that by 2030 or so, they will be able to put orders of magnitude more tonnage onto orbit.
China noticed - and it is trying their darndest to close the gap.
But spending on development of new steam engines when the competition already has electric ones is pretty backward.
https://en.wikipedia.org/wiki/RD-180
https://en.wikipedia.org/wiki/National_Security_Space_Launch...
https://news.ycombinator.com/item?id=30542226 ("Russia halts deliveries of rocket engines to the U.S. (reuters.com)")
(It's not a national security payload, but the Starliner ISS astronauts also went up on this Russian engine).
The success of this model hasn’t gone unnoticed by China, who are funding several private rocket companies (such as Space Pioneer) to develop reusable launch platforms in competition with the state-owned contractor CASC. They are making rapid progress!
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.
SpaceX is absolutely nothing new in the process, other than having been funded by explicit military program to prop up new space launch vendors.