This may indeed work, and it has been discussed before:
But I’m not aware that anyone is seriously looking into it, at least not currently.
Yes, but batch verification does not mean that the work of verifying n Schnorr signatures is the same (or similar) to the work of verifying a single Schnorr signature. Batch verification of n signatures will yield nice speed-ups as compared to verifying the n signatures individually, but it’s at most a factor of ~2 in practice.