Points mean infinite precision and infinite information for continuous real-valued coordinates.
These infinities just cannot exist.
The need for renormalization is telling us we are wrong. We cannot remove renormalization and leave the bare infinities, we have to change the substrate.
QFT is all waves, no particles. Particle localization is just Fourier Analysis over those fields. Heisenberg's Uncertainty Principle is just basic Fourier Analysis on wave packets.
QFT says fields propagate as waves, but interact as particles. A particle is just an interaction event, not implying a continuous localized existence between interactions.
But QFT is formulated over a pre-existing background spacetime, which GR denies.
The naive combination of QFT and GR just does not make any sense at all. We need a new theory of QG.
My hunch (prejudice perhaps) is that QG will be discrete. No points. No infinities.
A theory of quantum gravity won't eliminate the fact that electrons act that way when actually observed (they have a "smeared out" effect in a region near but outside of the nucleus that has physical dimensions greater than zero).
Why couldn’t a theory of quantum gravity remove the need for such renormalization without making electrons not point particles?