First and foremost: I'm glad that you enjoyed your time in Israel. Look me up on your next trip!
Again, I think that the easiest way to think about Judaism is as a mix of nation and religion. You can't have one without the other, although for as long as Jews have been around (including Biblical times), people were pushing the limits on both fronts.
There have always been non-Jews living among Jews, who are sorta kinda included in the Jewish people but not completely. And then you have Jews who don't observe the religion, who are only really considered kicked out if they actively practice another religion. And even then, if they renounce the new religion and return to Judaism, they're welcomed back.
Why is Judaism based on the mother, rather than the father? Because 2,000 years ago, that was an easy way to establish whether you belonged. If the rules were being made today, perhaps DNA testing would be the rule, but we're dealing with an ancient culture here that's trying to mix old rules with new realities.
Would a gay teen from Tel Aviv have anything in common with the haredi (ultra-Orthodox) from Bnai Brak? They might not agree on the interpretation of what it means to be Jewish, but I think that they would (perhaps reluctantly) see themselves as part of the same family. Just because your weird uncle is weird doesn't mean you disown him or stop talking to him; he's still part of your family.
Indeed, I stand by my assertion that Jews, wherever they live, feel like a part of a big family. But part of the family culture involves argument and debate, including over what is legitimately considered Jewish. Throughout Jewish history, there has been major, multi-lateral push-pull dynamic among different groups, as well as among different rabbis. And between popular sentiment and what rabbis allowed (or forbade). People are supposed to listen to rabbis, but they often don't. And of course, it's also common to say that you don't like a particular rabbi's ruling, but you have found a rabbi whose ruling is more to your liking.
Which is why you're not supposed to go rabbi shopping in Judaism. Which is why there's the famous saying, "If you don't ask the rabbi, then the chicken is kosher." (Meaning: Don't ask a question if you don't want to hear the answer.)
So Judaism is neither 100% people nor 100% religion. It's a mix of both, and you cannot separate them.
What has happened in Israel (where I live) is, I believe, a sad perversion of much of Jewish observance, in that you now have religious law handed political power -- a mix that's toxic to both. I'm convinced that one major reason why Israelis are so turned off to Judaism is that the people in charge have been so deaf to public interest and demands that people have turned off, or turned elsewhere. It's very sad for me to see so many Jewish Israelis think that Judaism is a primitive and stupid religion, when its very pluralistic nature means that people can and should find their own interpretations.
It's true that Jews have always been fiercely argumentative and divided, while simultaneously remaining loosely united. Much of the Jew-vs-Jew hatred that happens in Israel is an outgrowth of each group trying to impose its will, politically and legally, on the other. Suddenly, whether you drive on the Sabbath is no longer a personal decision, but a national one, which inherently leads to debate.
And the imposition of people's seating choices on airlines, and such? I believe that this is a similar outgrowth of political and religious power, which has convinced some that absolute adherence to their legal interpretation takes priority over being a decent person. Decency is supposed to be the #1 priority to someone who accepts Jewish law, and I see far too many examples of that not happening in Israel, I'm afraid.