Isaiah 53 is not referring to Israel. It is referring to the Messiah. Nowhere else in scripture is Israel called He. It is consistent to call Messiah He.
Jews have their point of view. Christians have their point of view. Literature can serve a multitude of different purposes and agendas depending on the reader.
If I was to try and please everybody as best I could, I would say this.
The Jewish people, individually and collectively, no matter how great and small, humble or proud, rich or poor, influential or invisible, are the embodiment of ancient Israel and Judaism, not just to themselves, but also to the rest of the world. Jewish Rabbis, the teachers and spiritual leaders of Judaism, are also the embodiment of ancient Israel and Judaism. Jesus was a Jew. He was also a Rabbi. As an individual, as well as a Rabbi, he was an individual embodiment of Judaism and ancient Israel to his own people as well as to the rest of the world.
Christians regard Jesus as "King of the Jews." That is the title they give him. Titles are just that: they are labels that only have meaning when people understand their context and significance. A king is not a king unless people treat him like one. It is more of a historical tradition whose significance has faded with time due to the widening gap between Judaism and Christianity. Jesus was a hero to the first-century Christians, a hero to the Christians among the Jews. To the Christians, therefore, Jesus was the ideal representative (and therefore "king") of the Jews.
Christians believe Isaiah 53 refers to Jesus because people condemned him and had him crucified, because he was persecuted and ended up being a hero. Jews believe Isaiah 53 refers to Israel. Considering the centuries of oppression and persecution that the Jewish people have endured, they too, have undergone tremendous persecution and oppression. Jesus' persecution ended with the crucifixion; for the Jews it was the Holocaust.
The question of whether you would regard Jesus as "King of the Jews" is a matter of perspective. It's a question of which Jew, throughout history has had the most influence amongst non-Jews. If you assess his influence according to the number of adherents of non-Jewish Abrahamic faiths (ie. Christianity and Islam), you would get a number that is a significant fraction of the world's population.
A king often has to serve as ambassador and diplomat. According to the numerical populations of Christians, I would say Jesus pulled off his "diplomacy" quite well. According to the above criteria, therefore, Jesus has a strong case of being an individual embodiment of Judaism (other than the collective embodiment), especially someone who has been influential posthumously among non-Jews.
So what does this have to do with Isaiah 53? Well, if Isaiah 53 refers to the embodiment of Judaism, to ancient Israel, to the Jewish collective, and Jesus is an embodiment/representative of Judaism, it could include him as well, especially as a representative of Judaism.
What I mean is, Isaiah 53 could refer to both.
Why do you think so many are intrigued by the knowledge of discovering that originally it was not what it is today? That Jesus was Jewish?
I think people are just interested in the role Judaism plays in the world. It doesn't necessarily result or lead to conversion, it may be just a way of giving credit to other religions.
Having said that, Jesus may indeed have been a Jew and a Rabbi, but whatever he said and did, he was not necessarily doing it for Judaism, but the world as a whole. If Jesus' life and sayings were of little value to Judaism, it's probably because contributing to Judaism wasn't his number one concern when he was doing what he was doing.
Jesus doesn't and didn't really need recognition in Judaism because he is and was already receiving recognition in Christianity and Islam. Because Judaism could not, did not or was reluctant (whichever one you like) to contribute a concept of Jesus that was different to that already offered in Christianity and Islam, it was natural that Jesus disappeared from Judaism.
Modern Jews who take an interest in Jesus aren't necessarily doing it because they aren't satisfied with their Jewish experience, but because they are interested in belief systems outside of Judaism. It's their way of making a contribution to Judaism: by seeing what role Judaism might play in the world at large, and the kinds of relationships Judaism might have with other religions.
Modern Jews who give credit to Jesus and Christianity are really giving credit to Judaism itself because Christianity came into existence through Judaism, through a Jew, so they are really taking an interest in a Jew who made a contribution to the world outside of Judaism, whether that specifically interests them or not.
True, a lot of credit has been given to Christianity for Western civilisation, but I think whatever credit it gets belongs more to Judaism. It could be thought of as how one Jew changed the world.
Was Paul the apostle (Sha'ul) a Rabbi? We know he was a Pharisee. We know he was asked repeatedly to speak in the synogogue also.
The same with Paul. He had an interest in making a contribution to the world outside of Judaism.