High voltage won't work on third rails because of the close clearances between the energized parts and between the grounded parts. There would be arcing from energized pieces to grounded pieces. There would always be arcing in any kind of wet weather because of highly conductive dirty water thrown all over by the wheels. B.A.R.T. has problems with this, as it is, with their voltage of 1000 volts.

All third rails make track maintenance difficult and more costly.

Yes, there are NO advantages, except scenic, to third rails!