Why do we have so few neutrinos to study?

It is because the neutrinos cross section is very small.

Roughly, the cross section is a measure of the interaction probability (for a given force) of the given particle with your target.

Since the probability for interaction depends upon the strength and range of the interaction, the cross-sections for neutrino interactions are very small since they interact only by the weak interaction.

Hence, it is not that we receive few neutrinos, rather it is our (in)ability to detect them, due to our current knowledge of their intrinsic nature, which prevent us from measuring high fluxes of neutrinos.

For this very reason, neutrinos detectors are made of huge volume targets, in order to maximize the detection probability.