Occasionally, American Christians will point to how Israel wanting a King in 1 Samuel is not a portrayed as a positive thing. But then equally American Christians like Chuck Missler will point to how Jacob was promised Kings would come out of him, and to instructions for Kings in The Torah to say that is taken out of the grander context of The Bible and argue the problem was only they couldn't wait for David.
The problem is that the demand for a King is equated with rejecting God as King. And whenever Israel wants to do what the other nations are it is always presented as bad. Monarchy in principle also seems to be vilified in the Abimelech narrative.
God knew this would inevitably happen, and so He planned for the bloodline He was going to incarnate into to become Israel's Royal Family. But Israel wanting a human King is still inherently a rejection of God.
Throughout the Book of Judges the narrative voice often says "and there was no King in the Land" which gets interpreted by Royalists as like the "everyone did what was right in their own eyes" comment as an explanation for why Israel kept falling into Sin and Idolatry during this period. Again that is consistent with their not respecting God as their King being the problem.
Instructions for The King are in The Torah, but not till Deuteronomy, which otherwise mostly only repeats or expands on things explained earlier. This suggests it was a very low priority.
The references to the Messianic Era in Ezekiel 40-48 and Zechariah 14 define it as The LORD ruling Israel as King again. Which I consider Old Testament proof that the Son of David would have to be God in The Flesh.
Shiloh was where The Ark and The Tabernacle were during the Judges period. Ezekiel 40-48's geography I feel places the vicinity of Shiloh as being where The Temple will be during The Messianic Kingdom. I think that is a good symbolic suggestion that in-spite of all that period's problems the Judges system is the ideal form of Human Government God intended for his people.
In Ezekiel The LORD is King and there is a "Prince" but the word for "Prince" here isn't Sar or Nagyim either of which could be viewed as a royal title, it's Nasi. Nasi is the title the Hasmonean leaders used originally before they started calling themselves Kings with the sons of Hyrcanus. It's also the term used to refer to the head of the Sanhedrin. It seems Nasi was the title used by the Exilarchs often as well. It's been suggested it's most accurate English translation would actually be President. Interestingly Ezekiel 34-37 seem to identify this Nasi as David himself resurrected.
Bible skeptics usually don't consider the NT era Sanhedrin to be a descendant of the council of 70 Elders mentioned in Numbers, but a development of Greek influence during the Hellenistic era. In Jesus time the Pharisees controlled the Sanhedrin and controlling that was the source of their power. Jesus said they sit in Moses seat, so He endorsed the notion of a line of descent.
The Sanhedrin in the NT is sometimes just called "the Elders of the people" if you search for similar references to "elders" in the Hebrew Bible you'll find their presence throughout Israel's history can be felt. Exactly how it was organized and manifested may have changed, but it was always there.
Monarchy isn't the only from of government that can go bad, and it's possible for an Absolute Monarch to be pretty good. But over all it is very risky to give any one man that kind of power.
Reason why I feel the false perception of Biblical support for Human Monarchy can be dangerous among American and politically non Monarchist Christians is I feel it can overlap with the desire to give the office of "Pastor" too much power.
Jerry Fawell said he wouldn't run for President because he wouldn't accept the demotion. That reminded me of the Justice League Unlimited episode were Luthor said to The Question "Do you realize how much power I'd have to GIVE UP to be President". Saying something like that is a sign of a megalomaniac.