But in other quantum field theories, there are vertices with more edges. In QCD, for example, there are vertices where 4 gluons meet. In the Standard Model, there are vertices where 4 Higgs particles meet, and vertices where 4 electroweak bosons meet, and so on.

The numbers 3 & 4 aren't really special. What's special is the numbers we assign to the edges. In 4 spacetime dimensions, any gauge boson (like the photon or gluon) gets the number 1. So does any scalar field like the Higgs. Fermions like electrons and quarks get the number 3/2.

These numbers are important because you should never meet a Feynman diagram in a particle physics computation where the sum of these numbers at any vertex is larger than the spacetime dimension 4. This is a fairly deep fact about physics. Diagrams that violate these conditions are strongly suppressed at "low" energies (such as those accessible at the LHC). They contribute almost nothing to the Feynman diagram sum.

Edit: This is Wilson's explanation of renormalizability, adapted to Feynman sums. It's the explanation for why physics is possible: The details of short distance physics (think Planck scale) can be very complicated, maybe involving lots of complicated Feynman diagrams. But at low energies, almost all of that complication averages out, and we can adequately approximate the laws of physics using simple ingredients.