I read in Urs Schreiber's notes on mathematical QFT that the infinities in the standard approach to QFT appear because the product between operator-valued field distributions is not always well defined. Especially in the standard definition for the time ordered product this gives problems (details).
I wondered why the fields are distributions and found this answer, stating that (already in classical field theory) the point evaluation functionals need to be smeared out for their Peirls-Poisson bracket to be well defined (and therefore become distributions). I guess that is a good mathematical reason, but now my question:
How to motivate intuitively/physically that (quantum/classical) fields should be distributions?
With "intuitive" I mean something in the spirit of the following explanation (which I find a bit vague and don't understand properly):
The need for considering distributions comes from the non-trivial structure of the theory at very short length scale where fluctuations are very important. At short distances, functions are not sufficient to describe the field state, which is not smooth but rough, and distributions are necessary.
(from A hint of renormalization)