a secular leader being legitimate implies that the Church is not supposed to be the government, but rules the realm of god or the "spiritual" world.

No, it's much stupider than that. They take credit for "secular" leaders, too.

Romans 13:

Let everyone be subject to the governing authorities, for there is no authority except that which God has established. The authorities that exist have been established by God. 2 Consequently, whoever rebels against the authority is rebelling against what God has instituted, and those who do so will bring judgment on themselves. 3 For rulers hold no terror for those who do right, but for those who do wrong. Do you want to be free from fear of the one in authority? Then do what is right and you will be commended. 4 For the one in authority is God’s servant for your good. But if you do wrong, be afraid, for rulers do not bear the sword for no reason. They are God’s servants, agents of wrath to bring punishment on the wrongdoer. 5 Therefore, it is necessary to submit to the authorities, not only because of possible punishment but also as a matter of conscience.

It’s not rocket science. It’s so obvious you can’t actually answer the question.

So I ask again: what has any of this to do with patriarchy? Or perhaps it’s too complicated, I should ask instead “what as any of this got to do with with a society ruled by male heads of family units- usually at the clan level and usually the eldest male.”

You’re also imputing an awful lot into a non-answer. The question Jesus was asked, is “was it lawful to pay taxes to Rome”, and that’s a question that has no right answer. Or at least, no answer that doesn’t have people trying to kill him.

In the scripture, Jesus even recognizes this fact. He gave a non answer.

• Multiple currencies were in use (the coins minted by Tyr were thought to be particularly good as they were known to not debase their currency, which is why the temple only accepted that currency for temple tributes and offerings.).
• taxes were not always paid in currency. They could also be paid in goods (especially crops or food,) and useful services.
• the coins were owned by whoever owned them, not the emperor. The seal and face of which ever emperor was on the throne only represents the authority to mint the coin, not ownership.

And again if he gave anything but a non answer, he’d have been killed. Either by the Romans or the Jews. Which doesn’t matter… except to say the Jews probably would not have been as torturous about it.

Finally, the concept of paying taxes to the lawful government and tributes to the temple being two different obligations far predated Christianity- probably also Judaism, for that matter.

The question wasn’t about “are secular governments allowed to collect taxes” or even what rates. Ultimately, the question was “is Herod Antipas (the then-king of Judah, who was a client of Rome) legitimate,” (because guess who was paying taxes to Rome.)(also take a guess at who was making a claim to Herod’s throne.)(hint: the messiah was supposed to be a king of line of David…)