The sun is the biggest neutrino source in the sky. So we'd need some way to filter out neutrinos that are not from the sun, like we do with various photon telescopes.
Furthermore, we can barely detect neutrinos. Building neutrino detectors is extremely challenging. Usually they are extremely massive and surrounded by lots of rock (even more massive). We'd have to get all that mass to the focal point of the observatory which is extremely far away.
Lastly, the gravitational field inside the sun is much different than outside. In fact, the field is strongest at the surface (or slightly below, as it doesn't have equal density). The further inside you go, the more parts of the sun start pulling you outside, until you reach the center of mass, where the gravitational forces cancel out.