Or walking... Do you walk a quarter mile or do you ride the bus? If the bus is free the number of able bodied people using the bus for trivial distances could increase.
Above a certain frequency buses become extremely unreliable and dwell times start impacting travel time because bus stops only have so much room. With trains you have more room to work with, but once you run out the marginal cost of building new lines runs in the hundreds of millions, if not billions of dollars.
When services are made free, people have to wait longer, paying with their time instead of with money.
People who don't want to wait and are not given the choice of paying for faster service end up having to buy cars instead, making traffic worse for everyone.
I don't think you can write off the decrease in foot traffic free public transit would cause.
Metro rail is sexy, but it's only one part of the transport mix.
I think that making public transport free in an easy step on the way to a city that relies a lot less on cars. Of course making it more attractive to walk or cycle and more expensive to drive is very important too, but that requires a lot more work.
it’s dangerous because roads are very narrow ( so protected bike lanes are rare) and there’s a lot of traffic.
the bike city service got changed recently to include electric bikes and as a result nothing worked for months ( not sure it’s back to normal yet).
Even if you manage to get a bike there’s a very high chance your bike will have defects because people like to do stupid things with them, like getting down stairs, or hitting the curbs.
and finally, if you think about buying yourself a bike, expect to have it stolen on average once a year ( depending on its price).
Very few people would have this luxury in city like Paris, where housing is prohibitely expensive in the city center, where many offices are traditionally located.
And, before you say "You might be inclined to find a job where more affordable housing is available", many people do not have so much choice in terms of where to work (same as working from home).
There is little point in making public transport free there. This is still money to be paid one way or another.