Electron falling into proton approaches infinite kinetic energy why?

You are stuck in the classical framework. It is an experimental fact that once the existence of electrons and nuclei was established by observations, and atomic spectra were observed, no classical electromagnetic solution could fit the data. So the answer to why the electron does not fall on the proton and reach the infinite energies that the 1/r classical potential predicts is: because that is what has been experimentally observed.

Atomic spectra , together with the photoelectric effect and black body radiation necessitated the introduction of a new mathematical theory that can fit the data and predict generally the behavior of atoms and molecules and much more. It is called quantum mechanics.

There are no orbits or trajectories for particles at the microlevel of electrons and nuclei, there are only probabiliy distributions, called orbitals, and the solutions of the equations that give these orbitals do not allow reaching the r=0 of the 1/r potential in the classical sense. There is a probability for an electron to pass through r=0 but that is all. There is no 1/r there. It has been used to calculate the orbitals.

I have a number of links in my answer to a similar question here.

Edit after comments:


look at the evaluation of the electron orbitals for the hydrogen atom in the x,y plane.

The s orbitals pass through r=0. ( zero angular momentum quantum number) At the same time the corresponding orbital of the proton ( let us not forget the nucleus) will also look like this but for a much, orders of magnitude smaller dimension. The combined probability of overlap is smalland the energy of the s state is not enough to make a neutron. Thus in the case of the electron and the proton this can happen in neutron star conditions, energy supplied by the gravitational field. Electron capture does happen with heavy nuclei where there is the available energy to change a proton into a neutron, with quantum mechanical evaluated probabilities.

Classically it's the same reason the earth doesn't fall into the sun or the moon into the earth. The equation of motion of a mass attracted to a central attractor is an orbit. So pre-quantum mechanics, physicists wouldn't be surprised by the model of electrons orbiting a positively charged nucleus. Now even suppose that sometimes an electron did crash into a nucleus and change a proton to a neutron. How would you know it had happened? In the case of hydrogen, the neutron would disappear, you'd have one atom less, and only hydrogen stable orbits would remain. So there is almost an anthropic principle there: Hydrogen is stable and what's not stable is not hydrogen! But now comes quantum theory, and it had better predict the already observed behavior, hopefully even more accurately, otherwise the theory would not have survived.