Why is mass renormalization insufficient to explain electron mass? In the Standard Model, I understand that the mass of the electron is assume to arise from two effects:


*

*A bare mass given by Yukawa interaction with the Higgs field, and

*A mass correction from mass renormalization effects
In this framework, why do we need to assume 1? Could mass renormalization explain the mass of fermions in general?
 A: In the actual local quantum field theories, theories of point-like particles, the mass correction due to the renormalization effects from (2) is divergent. It has a short-distance divergence so it is infinite. One needs to cancel the "infinite part" so that there's a finite leftover. What is the separation of the physical observed mass to (1) and (2) depends on various choices, "renormalization scheme", etc.
However, there's one more fundamental misconception implicitly included in your setup. In (2), you seem to assume that the loop effects only correct the mass but they don't affect the Yukawa coupling. But this assumption is invalid. The loop corrections affect the Yukawa coupling as well, so that the proportionality essentially holds before as well as after renormalization. So because we know that the physical mass of the electron is nonzero, we also know that the Yukawa coupling is nonzero. Both of them "run" i.e. depend on the renormalization scale but this running is logarithmically slow.
Note that the non-renormalized, bare parameters – whether we talk about masses or Yukawa or other couplings – are always infinite and only when the infinite counterterms are added, these infinite counterterms cancel the infinite part of the loop corrections coming from the bare parameters and we obtain a finite result. Again, this is true for electron's mass, Yukawa coupling, gauge couplings, and other couplings.
