I'm trying to avoid commenting on this thread because I think it is impossible for me to write a nuanced enough response that will give people a closer understanding of what it's like to live here (I moved here 10 years ago, though I spent 2 years in the UK, so 8 years total).
However, I can comment on the "Japanese only" signs. I've never actually seen one myself, but I know they exist in some places. An acquaintance of mine is well connected with a particular sumo heya (a "heya" is essentially a group of sumo wrestlers who live and train together along with their trainers and staff). It is popular (and fun) to go and watch the wrestlers train in their heya, but one is sometimes confronted with the "Japanese Only" sign at the entrance.
My acquaintance is not Japanese (he isn't even a resident), but like I said, he is well acquainted with the heya. One time, he asked some visiting foreigners if they wanted to watch the training and they were delighted to accept. When they got to the entrance they saw the sign. "Is it really OK?", they asked.
"It's fine. I've made arrangements. As far as the heya is concerned, if you know how to make arrangements to visit the heya, then you are Japanese".
Living in Japan can be very difficult for people who are not culturally Japanese. There is a weird inside/outside culture where either your are inside with a bunch of strange rules that you have to follow (and know to follow, somehow) or you are outside (where you don't really need to know or follow any rules). What "Japanese Only" means is that they only want people who know what the rules are, know how to follow the rules, and most importantly, are willing to follow the rules to the letter even if they think the rules are stupid. If you can do that, then you are "Japanese enough".
Generally speaking, there are few cultures in the world with the same kind of byzantine rule following culture that Japan has. Many cultures are literally morally opposed to blindly following rules. People from those cultures often have great difficulty in certain circumstances in Japan. However, if you show that you are willing and able to surmount those hurdles, my experience has been that most people have no trouble accepting you as Japanese.