If there's an increase in utilization of public transport, to the point where they need to run each route more often, or introduce additional routes, that'll be seen later on, when the impact of the ticket is evident. At that point, it can be weighed against any changes in private transport behavior.
It's probably premature to take upfront action against overutilization of public transport.
Then increase train fare so it’s at least as expensive per-carbon-emitted as driving. Since trains emit much less carbon, it will end up being cheaper to get from point A to point B by train.
Driving or taking the train has another cost, congestion. (more drivers means it takes everyone longer to get to where they’re going, more riders means some people might not be able to fit on the train. that second one used to be a common occurrence to me.)
Now that might seem hideously unfair to poor people. The solution is to take all the revenue collected this way and redistribute it throughout the jurisdiction where the law is enforced. The poorest would actually see an increase in their purchasing power. That’s because poor people emit less carbon on average, so if e.g. a poor person emits 1 ton per day and the average is 2 tons per day (making up numbers), they would pay the tax cost of emitting 1 ton but receive the tax cost of emitting two tons. The money would come from wealthy people who emit 3 tons but only receive the cost of emitting 2. That way everyone is incentivized to emit less without it unfairly burdening the poor.
At the end of the day the total number of trips are probably not very elastic, because the price (if it isn't a very high one) usually isn't even a consideration if you want to go somewhere (if it's not that far away anyways).
In fact, having lived both ways, I'd say you're far more incentivized to do this in a suburban setting with a car, since you're sitting in a private bubble with your own music and climate for every part of the trip except the store itself.
I think this really depends of the network you are talking about. In some European cities, public transit networks are a blast to use. If frequencies are high enough, and modalities are well connected, you can "ride" the city as easily and naturally on a metro/tram/bus as you would have done by walking. I've took a lot (a damn lot) of bus/trams just because they were there, passing by, and allowed me to save 3 or 4 minutes (with my annual ticket, price was never my mind).
(But I'm lucky enough to live in one of the lowest CO2 emitting countries in terms of energy mix.)
I do remember an UberPool trip I took. We got another passenger along the way. The driver ended up having to go 8 blocks out of our trip's path to pick up the passenger. She went 1 block and then got out. At least when you take a bus or tram out of convenience you are not being so wasteful.
Aside: for "unnecessary" trips, I'm setting aside the possibility of other options, such as "I already own a car and will now use that instead of paying 1 EUR".
NOW you are correct though, better for all would be for people to walk to their local shops and stuff and only grab transit when they need to go beyond the 2-4 mile radius that is considered walkable, and that would pull some of the public transit off the road/rails.
At least in Prague, they put more buses on the line, use larger buses, use streetcars with two cars, and they even build new underground lines (very expensive) in directions where the capacity of buses is not enough. So no, by hopping on a bus "that is already going", you are contributing to this.
We're trying to incentivize public over private transit because the former emits less carbon (in general).
However, public transit still costs carbon, so taking less trips total seems to be better for the environment (the "reduce" part of "reduce, reuse, recycle" - kind of). If the marginal cost to a public transit user per trip is zero, then that incentivizes taking small, useless trips (e.g. "I could wait until the weekend to pick up a few small groceries, but transit is free, so why not go now").
Wouldn't it be better to add a *small" fee per trip, e.g. 1 EUR (much smaller than private transit), to incentivize people to make fewer trips in the first place?