The question OP is proposing is linked to the question of the mass formulas. Here, what really matters is if the mass of the u quark is indeed very near zero and if one has some compelling theoretical reason to believe this.
The strong CP problem could not be of much help here as pointed out in the Dine's review. The reason is quite simple: If one should have a $\theta$ term into QCD Lagrangian, the neutron would have a measurable electric dipole. From experiments we know that is not the case and a lower bound is fixed. But the electric dipole of the neutron does not depend only from the mass of the quark u and so, having $m_u\approx 0$ is a sufficient condition but not necessarily the right one.
From a theoretical stand point, from QCD sum rules a lower bound for the masses of u and d quarks can be estimated. The main reference is S. Narison, QCD as a Theory of Hadrons (Cambridge University Press, 2007). I report here the estimation given in this book for the sake of completeness (chapter 53 in the book):
$$(m_u+m_d)(2\ GeV)>7\ MeV.$$
This grants a small but yet finite mass and whatever mass formula should satisfy this bound. Of course, this is consistent with $m_u\approx 0$. But a more recent review (see here) gives $m_u\approx 3\ MeV$ that is not so small but it is on the strong interaction scale. Smallness of $m_u$ and $m_d$ masses makes chiral symmetry a very good yet approximate symmetry.