Why don't fundamental particles propagate outwards like classical waves? If particles such as electrons are vibrations in a quantum field, why doesn't their disturbance propagate like one?
I just don't quite understand why the elevated energy in that spot is pretty much self-contained.
Thinking of it as a vibration or wave innately would entail the idea of it spreading out evenly over time like ripples in a pond, wouldn't it?
 A: One has to keep in mind that particles obey quantum mechanics, i.e., the only predictions that can be made are probabilistic ones. There are no predictions for individual particle interactions, only the probability of a particle interacting.
Quantum field theory (QFT) is a mathematical tool that can be used in different physical cases to describe interactions.

If particles such as electrons are vibrations in a quantum field, why doesn't their disturbance propagate like one?

This is a misunderstanding of the following QFT model used in particle physics to fit and predict crossections, decays, etc.:
There exists a quantum field theoretical model where interactions between particles found experimentally, see the table, are modeled in the following way for calculation using Feynman diagrams:

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*Every particle in the table is a field uniform in space time represented by the plane wave function of the corresponding quantum mechanical equation that describes it, Dirac for electrons, Klein–Gordon for bosons, etc.
These are the fields, and they are like a coordinate system, unchangeable, so no motion as you envisage can occur. A simple example how coordinate systems cannot oscillate,  water waves: define a coordinate system on which the motion of the water molecules can be mapped and predicted. Do the coordinates oscillate?


*On these fields differential creation and annihilation operators operate to create or annihilate a particle. An individual particle can at best propagate with these operators, and no collective field disturbance can happen with one particle.
A: As stated by several people including @Ruslan, @Marco Ocram, and @John Doty in the comments, this question is related to the following classic paper by Mott:
https://royalsocietypublishing.org/doi/10.1098/rspa.1929.0205
The question Mott asked is: if a particle's wave function spreads out in a spherically symmetric way, then why are particles always observed as straight line tracks in cloud chambers? Here is an example of particle tracks in a cloud chamber, just to illustrate the point (source):

What Mott realized is that the full quantum state is not a function of the particle's position only. The full state gives the probability amplitude for the particle to be in a certain location and the detector to have lit up at that location. So the full state evolves into a superposition of states, where in each state in the superposition the particle travels in a straight line in one direction and interacts with the bubble chamber in that direction, causing a straight line in the direction of motion of the electron. The full wavefunction of the electron + detector is spherically symmetric, but in each individual state in the superposition, there is only a particle track in one direction. We observe the detector in one of these states, and so always see one particle associated with a single straight line track.
A: Let us consider light waves as an example:

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*The classical photon field, or the electromagnetic wave, is realized when we deal with a huge number of photons. The statistics that photons (bosons) obey, called Bose-Einstein statistics allows for such a large confluence of photons since any number of bosons can be in the same state. In this instance, they behave like classical waves.


*Electrons, which are fermions, obey Fermi-Dirac statistics, so the large particle limit we can have for photons will not work for electrons, since the Pauli exclusion principle applies and we can have only one fermion per quantum state.
In case you may be thinking about the other bosons in the standard model, we can also describe these bosons by classical fields in the limit of large particle numbers. For example, the gluons, the $W^\pm$ and $Z$ bosons. But whether this can be done in the real world, I'm not sure since simply put, these particles operate at very short ranges and have very short lifetimes.
A: One should distinguish the wave function from the particles it describes. The particles are point, or close to point particles. The wave functions propagate outward and with them the probability of finding particle charge, energy-momentum, etc.
