Exactly. Messiah is the Hebrew word that means "anointed one" and is literally used in Hebrew texts to mean "anointed King".
It means the Messiah is anointed by god to be the next King of Israel.
The Greek word for "anointed" is Christos. Given the NT books were written in Greek, this is where the name Jesus Christ comes from. Jesus is the Greek translation of his Hebrew name "Yeshua" and it literally means "Jesus anointed", meaning Jesus is anointed by god to be the next King of Israel. Actually his name was Iesus until the letter J was invented around 500 years ago. Prior to the invention of the letter J, the letter I was like Y, in that sometimes it's a vowel and sometimes it's a consonant. J was invented to become the consonant use of the letter I.
So yes, everything you say about the 1st century, Jesus is political is 100% true. His followers followed him because they believed he was anointed to the be the next King of Israel. They also believed he would return Israel to the correct observance of the Torah, reunite the 10 lost tribes, and rid the holy land of its current foreign occupation (Rome). That last bit is why Rome killed anyone who people believed to be the Messiah. And there were many of them that popped up around that time, because the book of Daniel had a prophecy that around that time the messiah would arrive. So messiah claims were popping up everywhere, and a lot of them were named "Jesus" because another OT prophecy claimed the messiah would be named Yeshua.
The specific Jesus we know of today popped up from the Essenes. The Essenes were a sect of Jews that were both pacifists and opposed to wealth, and were vastly opposed to other Jewish sects like the Pharisees and Sadducees who they believed had taken the nation of Israel so off course with their wrong religious beliefs. Their Yeshua Messiah (Jesus Christ) came from Nazareth, and thus their new sect of Judiasm coined the phrase Nazarenes (Nazarth + Essenes).
While this Nazareth Essene Jesus Messiah and his followers are the one we are most familiar with today, back when they existed, again they were just one of many sects of Jews who were making claims that someone was the Messiah (Christos.... i.e. anointed). And again, because this Messiah (Christos... i.e. anointed) was to rid the holy land of Roman occupation, these Messiah (Christos... anointed) believers were rebels and traitors in the eyes of Rome. They were called "Christians" and were persecuted because they wanted the holy land rid of Roman occupation. This is where Christian persecution came from. And it wasn't just Rome that did it. It was Jews as well. It was Jews that didn't believe that a messiah was present and didn't want to change the status quo. They may not like Roman occupation, but they saw that being better than what would come from war.
Eventually these rebellious Christians cause enough damage that Rome did go in and level Jerusalem, and either enslaved or scattered all Jews (not just the ones that believed in a Messiah). Years later this even happened to all of Judea. Israel as a nation existed no more. And all of these messiah believing sects exited no more either, because without a country, what good is an anointed king of it?
So why do we have christianity today? The one, and only one answer.... Paul.
Paul was one of the persecutors of christians. He was killing them... even the pacifist nazarene sect. He saw no difference between different sects of messiah believing Jews. If they believed a messiah was present, then they were rebels and were executed. Hence why Paul was there where Steven the nazarene was executed.
But something in Paul changed in regards to the nazarene sect. Were not exactly sure what and how, because even Paul's letters and the book of Acts disagree about how that process unfolded (his letters say he went to Arabia for three years and had visions of Jesus where Jesus taught him "his gospel" and even allowed him to ascend into Heaven for more visions and instruction. In Galatians he is adamant that "no man" taught him the gospel that he preaches, and it's 100% sourced directly from his visions of Jesus. He's even negative towards James, Peter and John in his letters, even insinuating that Jesus knew they were too stupid to understand the great mystery and that's why he didn't teach them it while he walked with them and had to wait to teach it to Paul after he died. The great mystey being that Jesus is not just a Jewish messiah, but a savior of the entire world. The Books of Acts takes a much different view of Paul, one that shows him respectful of the Jerusalem "pillars", and one that shows them working together. Given it was written well after Paul died, and by one of Paul's associates, it's pretty obvious that this view of Paul and his relationship with the original nazarenes is apologetic in nature, and must be taken with many grains of salt. Paul's opposition and pleading that he is telling the truth is far more likely to be closer to the truth, even despite him likely being very one sided, and the truth being closer to the middle.
While the NT books do provide some history of this nazarene sect, most, if not all, of the NT was written by Greek speaking gentiles, and not by Aramaic speaking Jews. And it's a fair assumption that most, of the books including the synoptic gospels were written by gentile believers of Paul's great mystery gospel, which is that Jesus is a savior of all for those that believe that he is god. The jews didn't believe that. Not even the messiah believing Jews. Not even the nazarene sect of Jews. Again, like you said, they believed their messiah was the next king of Israel. And when Jerusalem and Judea were no longer Israel, their beliefs all died off too. All that remained was the gentile version that Paul created from his great mystery reveled gospel.
It's those that believe Pauline christianity is what the followers of Jesus believed, that think Christianity isn't political. As Paul's great mystery essentially says there are no political boundries for his gospel. To the messiah beliving Jews, it was entirely political, and Jesus had nothing at all to do with Gentiles. Giving his message to them was like "feeding the dogs".