During a discussion at PF, I found the following interesting way to think of these quantities:
mH ~ √2 mZ
mt ~ 2 mZ
Hvev ~ 2 √2 mZ
mW ~ √7 / 3 mZ
The last one may look a little odd, but it allows us to approximate sin2 of the Weinberg angle as 2/9.
The impetus was a comment by @arivero in which he pointed out that a tHWZ mass estimate due to Hans de Vries implies
(mW2 - mH2) / (mZ2 - mt2) = 3/8
Now in many GUTs, at the GUT scale, we have that
mW2 / mZ2 = 3/8
So it's as if (mW2 - mH2) / (mZ2 - mt2) is almost invariant under renormalization group flow, with mH = mt = 0 at the GUT scale.
We could even speculate that my set of four approximations above is an infrared fixed point. (The approximations are not exact, but one could think of these as valid at tree level.)
Unfortunately I don't see how any of this makes sense in terms of Hans de Vries's original physical hypothesis.
Anyway, I find that the LC&P formula also works neatly using the four approximations. And I would remark again that mZ is very close to the standard model's μ parameter.