The question assumes that the Ikkar is independent of a generic Messianic age. Rather the Ikkar is about it being fundamental to believe in a Messianic age. The Ikkar is:
1) That the complete fulfillment of Torah and Mitzvos is the purpose of the world.
2) That G-d will ultimately make that happen through Moshiach.
3) To want that to happen.
If someone doesn't think that about Torah and Mitzvos, then they are fundamentally missing something, not really different than someone fully keeping Torah and Mitzvos but not really believing reward and punishment - it takes it that they are of no real consequence.
This is from an article from Rabbi Yoel Kahn in one of his articles from the series called Macheves HaChassidus which was printed regularly in Kfar Chabad Magazine over 20 years ago. I have a binding of a bunch of those articles that was made as a special printing back in the day, and the one I'm referring to is titled למה האמונה בביאת המשיח היא יסוד בדת? on page 132. Part of this book was re-typeset and reprinted in 2001 and is available here, but that is only volume one, which does not cover this article. Perhaps a volume 2 exists or is coming out (the Rabbi is still alive and actively writing).
On the specific question of why Moshiach and not without a king, that is a detail, not unlike why believing in Chabakkuk is a necessary part of the Ikkarim, even though Chabakkuk on his own isn't fundamental to Judaism. Similarly, although Moshiach as a king of specified linage is a specific detail of Messianic Redemption, it is the general concept he is tied to that makes it fundamental.
That last paragraph is not in the article I mentioned. However, he does explore the difference between what is fundamental and what is not. What is not fundamental, if someone mistakenly believes something (in my example above, it would be that there was no such prophet as Chabakkuk) out of ignorance, that is just ignorance, not a lacking in his faith. But if he doesn't believe in an Ikkar, then his faith is bad. (If memory serves, that distinction is from Rav Chaim Brisker).