And what about Japanese people in Japan, are they still trying to become part of a Japanese culture?
The short answer to this question is "yes".
Obviously, in some sense Japanese people define what is Japanese culture. However, it is also true that, today, all cultures are disappearing into a monolithic modernity, and, from that perspective, there are many Japanese people who perceive "Japanese culture" as something antiquated and slightly foreign that requires study to understand.
For example, based on my experience here in Kyoto, I have no doubt that Japanese people do not
see keikogi as simply training clothes. When they do training in non-budo of any kind, they wear western athletic wear indistinguishable from that seen in the US. Keikogi is for Japanese
Most of the Japanese people I have met in my dojo here are essentially the same types of people you would find in dojos in the US. I have never been to a US dojo where there is a corporate ladder-climber, for example, and I haven't seen one in a dojo here in Kyoto, either.
As someone else has noted, the answer to the original question depends a lot on what you mean by aikido. If you think aikido is just a self-defense system or is a truly spiritual undertaking, then obviously the cultural trappings are not essential. However, I don't think most people fall into either of these two categories.
As for me, I probably wouldn't study aikido if it were done in street clothes and without etiquette. Many elements of the world outside the dojo are not to my liking, so I see studying any budo as a sort of... what?... meditation? I can turn off certain parts of my brain in the dojo that I couldn't if we were all wearing shorts and t-shirts with corporate logos or identity slogans.
This makes it sound like I have a social problem of some kind, but I don't think so. I don't get huffy over corporate advertising, it just isn't natural to me. Although the keikogi and etiquette probably seemed artificial, superficial, or forced to most people, I find they make the training experience more natural and visceral.