A wave function does not collapse upon detection?

I think this question arises from a simple misunderstanding of what a wave function is. The wave function of a particle doesn't need to be "wavy". The description of a system in quantum mechanics is always given via its state-vector in the Hilbert space and that can always be translated to the wave function of the said system in a basis of your choice, e.g., the position basis or the momentum basis.

A wave function $\psi(x)$ of a particle in position basis simply gives you the probability amplitude of the particle at position $x$ which is a complex number, i.e., it gives you two bits of information:

• The magnitude gives you the probability (density) that you would find the particle in the vicinity of $x$ if you measure its position.
• The phase gives you the information that you'd need on top of the probability (density) to construct the wave function in some other basis, e.g., the momentum basis, so that you can calculate the probabilities (probability densities) associated with the measurement of its momentum.

So, the point is that there is always a wave function of a particle -- regardless of whether it is very localized and point-like or not.

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As to why wave functions are nonetheless called wave functions, I think it's a historic relic. There are two tangible historic reasons that resulted in this naming, I think:

• The position-basis wave function of a particle that has a definite momentum is $\sim e^{ipx}$ and it is actually wavy. These are the famous de Broglie matter waves.
• The time-evolution equation that all wave functions satisfy is called the Schrodinger wave-equation (because it was the equation that was followed by the de Broglie waves, I suppose). One should note that the Schrodinger equation is not exactly a wave-equation although it admits wave solutions. It's more like a diffusion equation with an imaginary diffusion coefficient.

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