I'm reading through some notes on integrate and fire models and it says that we assume that spike arrivals are random.
Why exactly can we make this assumption?
Is it because experiments haven't shown otherwise thus far?
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Assuming random spike arrivals is just a convention that is chosen, not an assumption of some natural system. It's not a property of integrate-and-fire models. Probably whoever chose this convention did so for some demonstration (teaching) purposes. It may also be chosen to model "neural noise" which is a shortcut for "all the other stuff going on in the brain besides that which we are studying". In that context, it's common to model the effect of a consistent incoming signal on a random background.
An equivalent might be to see how an electronic filter would respond to either an impulse or to white noise: you could learn something about the filter (such as its ability to attenuate certain frequencies, or a certain response pattern to an impulse) by looking at either of those responses.
Similarly, by feeding noisy/random input to a integrate-and-fire neuron, you can find how changing parameters of your model neuron changes its output properties.