Why do we know that an elementary particle is a point particle?

[A. Neumaier]

It is a stable infraparticle, which means that it has an additional mass degree of freedom, which behaves like an additional momentum dof.

Here is more on infraparticles. For more on the branch point of the electron propagator, see, e.g., section II of

• T. Appelquist and J. Carazzone, Infrared singularities and massive fields, Phys. Rev. D 11, 2856–2861 (1975).