It's not actually that hard. You fund them through general taxes rather than fares. Then how much you pay is proportional to how much money you make -- even a flat tax does at least that -- as opposed to the largely fixed amount that corresponds to the amount the average person has to move around in order to live an ordinary life, which is approximately a head tax.
There isn't space in the current design. That's the thing you spend money to fix. Build subways in high traffic areas -- the ones where there is currently congestion -- and make them completely free to encourage people to use them (and eliminate the administrative cost of fare collection to both riders and government). Build more dense housing near the subway stops so people are traveling fewer miles, removing traffic from the roads -- this one doesn't even cost money, just stop prohibiting people from doing it with zoning. Build pedestrian catwalks or tunnels in high traffic areas to prevent crossings from congesting the roads and road traffic from killing the pedestrians. And yes, you can even add more travel lanes -- it's not always the thing you need but it sometimes is.
You don't have to rate limit the resource when you actually build enough of it to satisfy the demand. There exist roads that aren't congested, the demand for them isn't infinite.
