It’ll encourage the use of public transport over private vehicles (provided there is a good public transit network present). Public transport has got better efficiencies, and if it can supplant individual transit to a good degree, that’s not a bad thing.

As far as ‘overuse’ goes: how many people do you know who just travel on public transport for the fun of it? Even in places where people can travel for a flat monthly fee, very few people spend any more time on public transport than they need to. I doubt that free public transport would substantially change that.