Why does the Larmor precession frequency depend on $vec{B}$ instead of $vec{H}$?

The reason is that $vec{B}$ is the fundamental field. Traditionally (and in Maxwell’s traditional notaion), this was not really appreciated, and was $vec{H}$ treated as if it were the fundamental one; however, this was incorrect.

The reason for the mistake was that $vec{H}$ was easier to measure, as its sources was the free (i.e., experimenter-controlled) current. However, the source of $vec{B}$ is the total current—including medium contributions from orbital and spin motion—which is ultimately more fundamental. An equivalent way to see this fact that $vec{B}$ is more fundamental is that the microscopic Maxwell’s equations are written entirely in terms of $vec{B}$ and $vec{E}$, the fundamental fields, while the macroscopic equations involve all $vec{B}$ and $vec{E}$, as well as the auxiliary fields $vec{D}$ and $vec{H}$. For a linear magnetic material, the whole point of using $vec{H}$ (which can be derived from the macroscopic Ampere’s Law) is to resum the “loop” you have outlined.

Finally, by virtue of its more fundamental nature, $vec{B}$ is what appears in the Lorentz Force Law, $vec{F}=qleft[vec{E}+left(vec{v}timesvec{B}right)right]$, and so $vec{B}$ is the magnetic field that determines the trajectories of charged particles. Thus it, not $vec{H}$, sets the period of motions such cyclotron orbits or Larmor precession.

In a classical experiment Rasetti proved that $mathbf{B}$ was the effective field within a magnetized matter. Here is the abstract:

The deflection of mesons in a magnetized ferromagnetic medium was investigated. A beam of mesons was made to pass through 9 cm of iron, and the resulting distribution of the beam was observed. Two arrangements were employed. In the first arrangement, the deflection due to the field caused a fraction of the mesons to hit a counter placed out of line with the others. An increase of sixty percent in the number of coincidences was recorded when the iron was magnetized. In the second arrangement, all the counters were arranged in line, and the deflection due to the field caused an eight percent decrease in the number of coincidences. These results are compared with theoretical predictions deduced from the known momentum spectrum of the mesons and from the geometry of the arrangement. The observed effects agree as well as can be expected with those calculated under the assumptions that the effective vector inside the ferromagnetic medium is the induction B, and that the number of low energy mesons is correctly given by the range-momentum relation.

Rasetti, "Deflections of mesons in magnetized iron," Phys. Rev. 66, 1–5, 1944; https://doi.org/10.1103/PhysRev.66.1