Many USA transit systems have such a low farebox recovery ratio that they may even save money by getting rid of the overhead of fare collection. Cash handling is not cheap, nor is maintaining fareboxes and electronic fare collection equipment across entire fleets.
Even SF Muni which is heavily used has only a 22% farebox recovery ratio.
Isn't most fare collection automated nowadays? There's still likely to be an "info" booth at big stations for out of towners.
It's only in above ground systems where you have to have people watch against fare-beaters. But even then the tickets are mostly electronic and self-serve.
I think parent was suggesting giving some free rides away, not getting rid of fares altogether. Getting rid of fare collection services and infrastructure does save on costs (often comparable to the amount collected, as you say), but once that's done, then demand would presumably increase and possibly outstrip supply. At which point you need to increase supply (expensive, and you're not getting anything back from it), or decrease demand (by charging people for a journey). Every transport system will have it's sweet spot, and some cities have different priorities that others.
It seems wasteful to subsidize, say, 80% of a transit system's costs with taxes, and only use fares as a means to keep it underutilized.
And in many areas of the USA, I doubt removing fares will push demand past capacity -- it's not the $100 - $200 month in fares that keeps most people in a car that may cost them $700/month or more. [1]
Even SF Muni which is heavily used has only a 22% farebox recovery ratio.