Of course it doesn't. The cost of public transport is a cost to society. It will be paid regardless of ticket fees. Hence the word 'public' in 'public transport'. The difference is that by making the users of public transport pay you give people a good reason not to use it when there are alternatives available. But those alternatives have a hard time competing with free.
https://en.wikipedia.org/wiki/Free_public_transport
It exists, and it works. And the best bit: all the money spent on the whole administration around tickets, making tickets, enforcing laws, gates and so on can be spent on improving the system.
> all the money spent
you mean all the money saved? The unspent money are only a factor in the year following the year when the non-spend started. After that, no one will remember that they used to spend to some big figure on all the things you listed and assume that the operating budget of the transport agency has always been the lower figure.
> It will be paid regardless of ticket fees.
This is unfortunately not the way many systems in the US are funded. Many have to balance their fares with the service provided because they are reliant on external political factors for their public funding.