I think the author of this article misses the point of the "public good" argument.
The idea here isn't that transit itself is a public good, which is non-excludable (people can get the benefit without paying for it) and non-rival (one person having it means someone else doesn't have it).
The idea is that the availability of quality transit is a public good. Who benefits from good public transportation? Well, if you own a shop downtown, you benefit because 1) employees can come to work for your shop 2) customers can come to buy things from your shop. If all the other shops decided to get together and subsidize public transit, you could just not join in and reap the benefits without paying the costs -- i.e., it's non-excludable. On the other hand, the fact that your neighboring shop's employees and customers are using public transit doesn't stop your employees and customers using public transit -- i.e., it's non-rival.
Quality public transit can help ensure that poorer people can get higher quality jobs. Who benefits when the poor can get quality jobs? Well, everyone -- there are more workers, there are more consumers, there's less crime, the hospitals are less overloaded, the businesses in the city center can afford to hire staff, etc. The city as a whole is a lot nicer, not just for the poor people, but for the rich people who can afford to fly around in helicopters. Again, if all the other rich people decided to band together and subsidize public transit so that their city would be nicer, you could just not join in and reap the benefits without paying the costs -- i.e., it's non-excludable. On the other hand, the fact that all your rich friends are enjoying a nicer city doesn't stop you from enjoying it as well -- i.e., it's non-rival.
The same argument goes for education. Obviously one person's education itself is a private good; but the availability of an educated workforce is a public good which benefits all companies; and the universal availability of education benefits all people: if everyone has an opportunity to better their circumstances, there's lower crime and so on; and you never know who may discover the next cure for cancer, or the next fantastic technological breakthrough.