Why do we not know whether or not neutrinos are their own antiparticles?

We know that neutrinos and antineutrinos exist, and it's possible to tell the difference between them. For example in a charged current detector electron neutrinos produce this reaction:

$$ \nu + n \to e^- + p $$

while electron antineutrinos produce this reaction:

$$ \bar{\nu} + p \to e^+ + n $$

And detectors can easily distinguish between electrons and positrons so they can easily tell neutrinos and antineutrinos apart.

In your experiment you emit a beam of neutrinos, and the detector would detect only neutrinos not antineutrinos, but that isn't proof the two are different particles. The neutrinos your detector emits all have left handed chirality while antineutrinos all have right handed chirality. You can change the chirality by having your detector move in the same direction as the emitted neutrinos at a speed faster than the neutrinos are travelling. That would mean the left handed neutrino in your frame would be right handed in the detector frame. If you did this and your detector started detecting antineutrinos you'd have proved the two are the same particle.

But for obvious reasons this is not a practical experiment. Because neutrinos are so light they travel at nearly the speed of light even when they have small kinetic energies. Designing an experiment where the detector moved faster than the neutrinos would be challenging at best!

Good point, such experiments have been done and they see nothing. But there's something more.

The neutrino source provides only left-handed neutrinos, their spin pointing against their direction of travel, because the weak interaction does that.

The detector can be sensitive only to right-handed antineutrinos, their spins pointing in their direction of travel, because the weak interaction does that.

We see nothing - but that could be just because the neutrinos have the wrong handedness, and not because of a neutrino/antineutrino difference.

So it's not proven, and we need the double-beta-decay experiments.