Quantum Field Theory lecture notes by David Tong (2007) puts it well:
In classical physics, the primary reason for introducing the concept of the field is to construct laws of Nature that are local. The old laws of Coulomb and Newton involve "action at a distance". This means that the force felt by an electron (or planet) changes immediately if a distant proton (or star) moves. This situation is philosophically unsatisfactory. More importantly, it is also experimentally wrong. The field theories of Maxwell and Einstein remedy the situation, with all interactions mediated in a local fashion by the field.This is also mentioned e.g. at Video "The Quantum Experiment that ALMOST broke Locality by The Science Asylum (2019)".
People have then tried to recover that by assuming that there is some inner sate beyond the Schrödinger equation, but these ideas are refuted by Bell test experiments, unless we give up the principle of locality, which feels more important, especially in special relativity, where faster-than-light implies time travel, which breaks causality even more dramatically.
If something does a quantum jump, what causes it to decide doing so at a particular time and not another? It is expected that a continuous cause would have continuous effects.