Isn't the "profitable public agency" you're looking for an oxymoron? Running in balance? Sure, but the word "profitable" makes me cringe. If there was profit to be found, private enterprise would've stepped in long ago.
There are profitable public transit agencies. Not in this [transit-backwards] country though. MTR in Hong Kong comes to mind. The public Japan Railways are profitable, as are the myriad private railways that operate closely together in Japan. And that's before you factor in the profits they make from real estate.
What makes our transit systems hemorrhage money is a combination of factors. First and foremost is land use policy. We have an insane fixation on building parking lots on valuable station adjacent land. Parking lots are the least effective way to build ridership, since you are limited by the relatively few number of spaces. The reason why we hurt ourselves this way is due to perverse Federal incentives as well as poorly thought out zoning codes which have proliferated across the country. In order to have profitable transit, you need heavy TWO-way ridership ALL day. That means there must be a variety of destinations at nearly every station on the line, so that people have reason to go there at all times of day (instead of the usual inbound/outbound pattern we focus on).
Second, we have fairly incompetent management of agencies and weird political divisions that get in the way. There is a major NIH factor at work which prevents us from importing best practices from other places. The fact is, Americans have lost the ability to plan, build and operate passenger railroads competently over the past generation or two. We need to get that back. But we're going to have to play catch up for a while. And Americans hate doing that -- it hurts our ego. So they'd rather pretend that everything is just fine, and we continue to run 50s-style commuter operations with lumbering, slow diesel locomotives, in the 21st century.
Finally, the main competition is heavily subsidized by the government: highways and roads. Railroads and public transit are high fixed cost businesses. In order to meet those costs, they need to attract as many fare-paying customers as possible. But if those customers are tempted away by freeways and under-priced gasoline, then the transit system gets screwed twice over. Hence, most have devolved into basket cases on life support, from which most any sane, competent upper management has fled for greener fields.