The nice thing is that even today, some American astronauts are flying to the ISS from Russia on Russian rockets, and some Russian astronauts are flying from the USA on American rockets. They're doing that for redundancy of getting their people to space, and for the general cooperation.

The place was originally selected because of the combination of several factors:

- Closeness to the equator.

- Ease of access by rail.

- Lack of populated/foreign areas at the path of falling booster stages.

- Relative closeness to production centers.

- Far enough from borders and small population around the site to make spying harder (remember that in the early days it was also an ICBM launch site).

Other central Asian Soviet republics were farther south, but much worse on the other factors.

Russia simply continues to use it because of the existing infrastructure and tries to gradually move its civilian space program to the new Vostochny Cosmodrome.

Airbus has significant (at least 25% last I looked) state ownership by various European governments. They did the development, the launching is done by Arianespace which is half owned by Safran (which is also part owned by France). That combined with the Ariane programs themselves being initiated and funded by the ESA makes it a much weaker claim to be a commercial rocket launch.

It's going to be hard to find a space rocket company which has 0% state involvement (e.g. Isar Aerospace and SpaceX have also accepted some amount of funding through venture rounds or contracts one way or the other at some point which can be traced back to governments, or purchased use of government launch facilities) by nature of "why wouldn't you accept a government launch at some point if you've already built a cost efficient rocket" or the like... but I think it's fair for Isar to claim to be launching the first commercial European space rocket since the program to build the rocket wasn't initiated by a state space agency or fulfilling a state agency contract by doing the initial flight.

Not only Airbus though. 74% of Arianespace is owned by ArianeGroup, which is 50% Airbus and 50% Safran.