I searched around a little; turns out that the system you propose is already in use on some lines in Europe. One is the Cologne-Frankfurt high-speed line (another one electrified with the 15 kV/16.7 Hz system), where the heating current is applied in bursts in the night hours; unfortunately, I couldn't find details (voltage, frequency if any, length of sections, method of shorting). Another is a rack railway in Switzerland, which runs on DC, but the heating current applied during the night is 50 Hz AC, and applied continuously.

However, neither system was found sufficient.  At the Swiss line, the problem is the time between switching off the heating and the transit of the first train. On the high-speed line, my (German) source doesn't say what's the problem, but says that the more conventional catenary brushing cars are in use, too.

(Other conventional methods I am aware of: spraying a de-icing fluid, sending out a loco in the morning with both pantographs up so that one scraps the ice while the other draws current, and using a 'flamethrower' to de-ice the worst spots.)

_{*Lunatic*, n.
One whose delusions are out of fashion.}