[Myrtonos]

The Bordeaux example has a street compatible form of third rail power on sections without wires. The particular system (Innorail/A.P.S) in Bordeaux would not work for buses, and even that system has some sections with overhead wires.
And indeed the systems with battery trams also have some electrified sections, so even these don't do without dynamic contact systems altogether.
If a light rail system or line is mostly off-street and street running sections are very short, and the vehicles are equipped with batteries, they could use battery power on street with off-street sections equipped with standard third-rails.

But any new trolleybus built these days would have auxiliary power units, these being important for temporary diversions. Trolleybus routes might not need to go through the most visually sensitive areas or if they need to go through them, run on battery power in those areas.