The United States is certainly happy to (launch military payloads with SpaceX). Having access to far more launch capacity, far more cheaply, is surely a significant strategic advantage.
Edit: Added text in parentheses for clarity.
Edit: ok, got it now. This is about a European rocket serving European strategic needs, why would I be talking about USA?
It's a huge fucking problem if you're not American. If you don't want to be subject to American law. If you don't want your technology inspected or tampered with by Americans. If you don't want the transaction to be in dollars (extra-territoriality of US law on all dollar transactions). If you don't want to deal with the problem of American citizens and companies subject to FISA, who could face severe repercussions in the US if they don't spy or facilitate spying on their non-American clients.
The US isn't really interested in the secrets of some Malaysian or Brazilian TV satellite.
Once again, compliance with US regulations, including export control, dual-use goods and tax regulations, can be weaponized, and has been in the past.
Maybe the TV satellite manufacturer has also supplied a weather satellite to Brazil that contains technologies covered by export control, bingo. Your satellite and your assets in the USA are frozen until the manufacturer provide all the information needed to decide whether or not it was a spy satellite.
The godamn Mars rover is covered by 14 export control technologies. https://www.bis.doc.gov/index.php/documents/pdfs/1008-satell...
It's a minefield and it was designed to be one. I know of several non-defense companies whose first question on the phone is "are you a US person as defined by FATCA and FISA ?".
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!
SpaceX is absolutely nothing new in the process, other than having been funded by explicit military program to prop up new space launch vendors.
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.
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).