First off, the right part of the graph is not really contributing anything: the $W$-boson, $Z$-boson, and Higgs boson all have mass in the same range, because these masses all come from the scale of electroweak symmetry breaking. Similarly, the left part isn't saying much, because everything is zero or too low to measure. So we're left with the $6$ quark and $3$ charged lepton flavors.
Explaining their masses is known as the flavor problem. Unfortunately, even though it really seems like there's something going on here, nobody has succeeded in writing down a compelling theory. Literally thousands of attempts have been made, but the resulting models generally are quite contrived, and either don't make any sharp and reasonably testable predictions, or make incorrect predictions. So most of these models don't attract much attention, for the same reason the Koide formula doesn't -- they're interesting, but it's not clear what more we can do with them.
A Taste of Flavor Model Building
People can spend decades working on this problem, and there's a great diversity of approaches, so I'll just give a taste. For a brief review, see here.
In the Standard Model, the quarks and leptons get mass through the Higgs mechanism. The symmetries of the Standard Model allow these particles to interact with the Higgs field, by emitting or absorbing one Higgs boson. When the Higgs field gets a vacuum expectation value, this turns into a particle mass.
The popular Froggatt-Nielsen mechanism is a twist on this. A new Higgs-like field $S$ is added and a new symmetry is postulated, so that different particles can interact with the $S$ field, but only by emitting or absorbing $n$ $S$ particles at once, where $n$ depends on the particle. The resulting mass scales as $\epsilon^n$ for some small parameter $\epsilon$, and fixing the $n$'s by defining the symmetry just right, you can reproduce the exponential structure in the quark masses. The quantitative agreement ends up about as good as your line. But then one could complain that this just reduces the question to why the symmetry has to be that way.
Another idea along these lines is radiative mass generation, where again the symmetries are fixed to give a hierarchical structure. Here, the heaviest generation (top, bottom, tau) can get masses at leading order in perturbation theory ("tree level"), but the second generation can only get masses at next-to-leading order ("one loop") and the first at the next order after that ("two loop"). Again, you need to set up the symmetries in a somewhat Rube Goldberg-esque way to make this happen.
In general, if you want to solve the flavor problem, then grand unification is useful, because it gives you a relation between quark and lepton masses in the same family. For example, in the simplest possible $SU(5)$ GUT, we have a relation between the positive and negatively charged quark masses in each generation,
$$\frac{m_b}{m_\tau} \approx \frac{m_s}{m_\mu} \approx \frac{m_d}{m_e} \approx 3.$$
This relation is not very accurate, but you can see how the rest of the reasoning would go: add some more Higgs fields to fix up the mass relations within each family, then combine it with one of the previous ideas to get an exponential hierarchy between families, and we've explained the pattern! The only problem is that in the process, we've introduced far more parameters than we managed to explain. Moreover, grand unification essentially always comes bundled with weak-scale supersymmetry, so many of these models have been rendered irrelevant by the LHC.
A completely different route is to appeal to the anthropic principle, or to cosmology. You can argue, for example, that the up and down quark masses can't be too far apart, or else you wouldn't get the rich structure of nuclear physics. Similarly, you can't adjust the electron mass too much without messing up the structure of chemistry. And the top quark, since it couples by far the strongest to the Higgs, determines the stability of the vacuum of our universe. But I'm not aware of any way to use these ideas to solve the whole flavor puzzle, because the other $8$ masses have essentially no impact on everyday life or cosmology.
The flavor problem is infamously hard. At the end of the day, we don't have anything that's essentially better, scientifically, than the line you drew. But lots of people are working on it!
