"international" as in "multiple nations", not as in "everyone". It's a cooperation between multiple nations. If the US won't take you up you can still ask Russia, the EU, Canada and Japan if one of them will host you on the ISS.
In general, the ISS is international. But each country has its limited resources that can be spent on getting people into space. Additionally, the ISS can only support so many people at once, and parts of the station belong to different countries (as I understand it). As far as I'm aware, nothing is stopping other space agencies from having a similar program.
I understand why people want even more progress, but getting private American astronauts into space to conduct research is a significant first step as far as I'm concerned.
If the ISS is actually international then shouldn't access be open to all.
Nor that one expects USA decision makers not to extend their imperialism to space though.