If overhead wires were the only advantage of light rail, why is anyone using light rail instead of trolleybuses?

The answer is of course that rails provide major advantages on their own, primarily by allowing vehicles to travel much more efficiently thanks to the low friction of running steel wheels on steel tracks

Efficiency — mainly through lower friction

“Thanks to the very low friction between the steel wheel and steel rail, railways can move a load using as little as 15% of the energy that road haulage needs.”

Gareth Dennis, How Railways will Fix the Future (https://www.penguinrandomhouse.com/books/761930/how-the-rail...)

It does make the name kinda ironic, since the batteries will make the tram heavier then a normal one that's powered by overhead wires, so its not 'very light'

There's various possible reasons. Less pollution from tire wear and the possibility to lay tracks across grass to create relatively ecological dedicated lines so the carriages don't get stuck in traffic are two options that spring to mind immediately.