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That was the old thinking. Turns out it leads to divergences which was one of the difficulties that quantum theory overcame. Baez has a good overview of these kinds of problems:

Struggles with the Continuum, https://arxiv.org/abs/1609.01421

Lots of issues spring from unphysical assumptions like continuous quantities, point-like objects and infinities. Quantum mechanics was the first step towards more discrete formalisms, and I think future physics will take further steps in this direction to eliminate these issues.

I believe the word you're looking for is quantum - the "point" still has a radius, a singularity does not.

An electron is a fluctuation of the electron field, not a singular point.

It certainly acts like a singular point if you measure it and have it scatter on stuff...

The electron wavelength is around 4 picometers at low energies. The wavelength also imposes a limit on the resolution of electron microscopy, for example. The wavelengths are really small, but they are not zero.

Sure, but when an electron hits a surface (like a detector), it doesn't hit it with a splotch of 4 picometer in size, it hits it as a dot (as far as one can measure, given the uncertainty principle)... right? Same as any photon.