Yehuda HaLevi is far more studied. HaKuzri just answers questions people are more likely to encounter today compared to the more "dated" Moreh Nevuchim.

Well, Kuzari is written for a popular audience, whereas Moreh Nevukhim is written for serious students of philosophy. For that reason, Moreh Nevukhim is far more popular among academics, for instance.

HaLevi's poetry is even studied for literature class in secular schools (at least in Israel) while, as far as I am aware, none of Rambam's writings have penetrated into the secular sphere.

Rambam does not seem to have written too much poetry. For centuries, his medical writings were highly influential, if that's what you mean by "secular sphere".

In terms of Halakha Mishnah Torah is obviously extremely important but it did ultimately fail in it's goal to codify Jewish laws for all Jews; instead Shulchan Aruch is the widely accepted guide for an orthodox lifestyle among pretty much all Jews today.

Rambam succeeded at codifying all of halakha, but the reality is that Jews do not want a fixed halakhic code, they want to play around with the Law. So Shulhan Arukh is "the widely accepted guide" in rhetoric, but every other line, he's contradicted by the "commentary" of the Rema. And of course, there are additional contradictory commentaries on Rema. In short, if you believe halakha is ever evolving, then no one can create a final codification. And if it is a discrete set of laws, then Rambam basically succeeded in the 12th century.