Arguments we have put before, but I may as well have another shot again.
A key upside of 3rd rail is that it is so easy, and cheap, to install. I saw the installation of it on Stratford-North Woolwich one weekend in the 1980s. I don't know if they did the whole lot in one Sunday, but it looked possible. One works train, ground crew, screw pots in, drop 3rd rail onto pots, on to the next length. Try that with 25kV, GWML lot. Notably it was a (onetime) Eastern Region crew from Chesterton Junction, Cambridge, pw yard doing it, so not a lot of background 3rd rail experience either.
Much is made of the additional lineside substations. But a 12-car overhead emu had three substations, one under each motor coach, and all the hoopla of pantographs, transformers, rectifiers, etc on board each one. Lineside substations are so straightforward compared to putting them on the rolling stock.