Why don't EPSPs travel passively through axons?
EPSPs decay over distance. The decay is described by the length constant, which can be calculated based on the electrical properties of some compartment (i.e., a length of dendrite or axon). By definition, the length constant is the distance at which the amplitude of the signal will decay by about 37%. If you are familiar with the concept of a "time constant," the math is exactly the same.
There are two main factors that influence the length constant: the number of open channels (i.e., how "leaky" the membrane is) and the diameter of the neurite. The "leakiness" of different parts of the cell is often modulated by inhibition. Preventing EPSPs from reaching the soma or decreasing their amplitude by increasing leak is called shunting inhibition. However, for your question, it is the diameter and overall length that is important.
Dendrites tend to be a bit bigger than axons, so they have a longer length constant: that is, EPSPs travel further. Dendrites are also not entirely passive: dendrites can actively propagate signals (see review by Yuste and Tank) using voltage-gated channels to boost EPSPs, especially in cells that have a long distal dendrite, such as the deep-layer pyramidal cells in neocortex. These voltage-gated channels give the dendrite a longer effective length constant
The most important thing is that even relatively short axons are just way too long for passive conduction: that's the reason for action potentials. Length constants depend greatly on the size of the neurite, but you can expect the distances to be on the order of millimeters to 10s of microns.
That said, it isn't ever true that a subthreshold EPSP won't flow into an axon: it just will get weaker and weaker with distance. Neurotransmitter release depends on a substantial depolarization which opens voltage-gated calcium channels; a small little EPSP that has decayed several-fold from the soma has no chance to cause calcium influx and subsequent vesicle release.
Spatial and temporal summation
EPSPs don't 'wait' anywhere, the EPSP is constantly decaying because all membranes are somewhat leaky, and current isn't restricted to any compartment. Current is constantly flowing out of the membrane and out into all the other processes of the neuron (and then out through those parts of membrane too!). In fact, if you are talking about passive conduction of EPSPs, you are really talking about a form of "leak", not through the membrane but through the cytosol! The EPSP isn't traveling to the soma in any directed way, there is current "leaking" in all directions, and some of it happens to go towards the soma. To get summation at the soma or anywhere else, EPSPs have to arrive both close enough in time and close enough in space.
EPSPs can sum with each other in the dendrites if they occur close enough in space, but of course that also only works if they occur around the same time. Similarly, at the soma, an EPSP slightly depolarizes the soma but only for a brief time. If one EPSP has an amplitude of, for example, 5 mV, and another EPSP has an amplitude of 3 mV, they will only sum to close to 8 mV if they occur simultaneously. If the 3 mV EPSP comes a few milliseconds later, the peak voltage might only be 6 mV, or if they are separated even more in time, there may be no summation at all.
Why do spikes start at the axon hillock?
The axon hillock is a special part of axon, right at the beginning of the axon. This area is more than just the first section of axon, it also typically has the lowest threshold for initiating an action potential due to a greater density of voltage-gated sodium channels. Sort of by definition, a cell's threshold is going to be the threshold for whatever part of that cell has the lowest threshold (at least near the soma, which is so big compared to axons/dendrites that it is nearly isopotential).
For example, take an example cell whose overall threshold is at -45 mV. If you were to somehow remove the axon hillock, leaving just the rest of the soma and the dendrites, the threshold wouldn't be -45 mV anymore, it would be much higher, maybe -10 mV. Even if you had some really huge EPSP (or manually injected some current) to bring the cell towards -10 mV, the axon hillock would still hit its threshold first and start the action potential well before the EPSP reaches a peak sufficient to start a spike in the soma proper.
If it helps you could think of a balloon analogy. The EPSP is air flowing into the balloon, and the action potential is the balloon popping. If one part of the balloon is thinned/weakened, that part will always be the site where the popping starts. It doesn't matter if you put in enough pressure to cause the thick part of the balloon to pop, because before you can possibly get it to that pressure the weak part has already burst.
Also note that for this analogy to fit with all the rest of this answer, it has to be a leaky balloon, so it only pops if you put enough air in fast enough before it has a chance to leak out.
Yuste, R., & Tank, D. W. (1996). Dendritic integration in mammalian neurons, a century after Cajal. Neuron, 16(4), 701-716.