First, one assumption:

Transit is economically beneficial; i.e., it creates more value than it destroys. (And set aside whether this is actually true.)

So, here’s an imaginary conversation between The Suburbanist and an enquiring interlocutor (EI):

Suburbanist: An unsubsidized transit fare might be about five dollars, so that’s what users should pay.

EI: That’s nuts. People can’t afford to pay five dollars for a bus ride!

Suburbanist: Sure they can. In fact, they are paying that right now through a combination of fares and taxes.

EI: But taxes are more equitable than fares alone because everyone benefits from transit, so therefore everyone should pay something.

Suburbanist: Everyone also benefits from transit users eating breakfast so they can have the energy to get to work. Should their breakfasts also be subsidized with taxes?

EI: To some extent, yes. That’s why we have food stamps.

Suburbanist: So a parallel policy would be means-tested transit vouchers, and not a general transfer of wealth to anyone who uses transit. (Of course, a less paternalistic policy would be means-tested cash grants to be used as the recipients choose.)

EI: But even if you can afford to pay five dollars, who would? That’s a lot of money! Transit would cease to exist.

Suburbanist: Transit is economically beneficial; there is money to be made by having it. Business are not going to walk away from profits. Salaries would adjust to reflect the shifting burden of paying for transit because real wages are set by supply and demand, and will therefore remain constant regardless of whom the government places the nominal burden of taxation on. It’s a good guess that lower employer taxes (and employer surplus created by transit) would be converted to employee wages. And that lower employee taxes (and employee surplus created by transit) would be converted to higher fares. In the end, those who benefit the most from transit will be paying the most -- instead of the current system where transit is paid for by profitable businesses (and passed along to their employees and customers) only because they happen to be profitable -- regardless of how much they actually benefit from transit.

EI: That still doesn’t seem right, because there will still be businesses who will free-ride on the costs paid by others.

Suburbanist: Not so, because transit is not a public good. The costs can be captured by fares, and then paid by beneficiaries as the actual burden of fare payment filters through the system.

EI: What about people who need transit for reasons other than going to work? How can they be expected to pay five dollars?

Suburbanist: 1) Fares do not need to be five dollars for everyone; transit agencies could practice price discrimination, as airlines do. 2) User taxes will be lower, so that will partially offset the fare. 3) In the longer term, and politics notwithstanding, competing transit companies would try to attract riders by lowering their costs and fares. 4) How can people be expected to pay five dollars for anything? Why single out transit? How do they pay five dollars for a pack of batteries or a pair of socks or whatever?

EI: But what about transit services that are not profitable? Who will pay for those?

Suburbanist: Are you saying that transit is not economically beneficial? Check that opening assumption again and ask yourself why it is essential to provide services that destroy value.

You sell 1,000 bagels every day for one dollar each. Then someone comes into your store and says that one dollar is an unaffordable price, and that you will be prohibited from selling any bagels at all unless you set aside 200 bagels to sell for fifty cents each.

Set aside the morality of someone threatening you for not conducting your affairs as they demand.

What happens next?

1. You could try to sell the fifty-cent bagels on a first-come first-served basis, but that isn't "fair" because that cheap-bagel line in front your store might include people who could afford dollar bagels. Why should they be able to buy cheap bagels? And besides, it's humiliating to make people wait in a line.
2. You could have a lottery in your store. That gets rid of the line, but it still has the pesky issue of people getting cheap bagels that they don't deserve.
3. You could make your customers submit financial statements so that you (and your auditors) ensure that only the most deserving people get cheap bagels. This assumes that it costs very little in time and money to prepare financial statements. But then a problem arises: People who don't even want fifty-cent bagels will buy them anyway, and then resell them for a dollar. Hang on! That defeats the purpose of paternalism. The goal is not to make others better off as they see fit; the idea is to make them better off as their (pretend) patrons see fit; i.e., with bagels.
4. After making your customers enter a lottery drawing and submit financial statements, you then make them sign contracts that prohibit them from reselling bagels for more than fifty cents. That ought to make them eat their own bagels, even if there are other people who want/need the bagels even more.
5. You finally have what seems to be a better idea: Sell 800 bagels for a dollar, and 200 stale bagels for fifty cents. If the bagels are sufficiently moldy, then you probably won't even need the lottery, the financial statements, and the resale restrictions -- but since the regulatory mechanism has now calcified, these restrictions remain.

So, every day, 200 moldy bagels are now sold to people who are willing to to endure lotteries, financial statements, and complicated legal restrictions.

Who gets the other 800 bagels?

1. The supply of fresh bagels has gone down from 1,000 to 800, and cannot be made up my making more fresh bagels because that would violate bagel-producing restrictions; i.e., the tradition/culture/character of bagels would be harmed if too many fresh bagels are made.
2. The price of the 800 fresh market-rate bagels is now $1.25. If you lost the lottery, now you can't even get a fresh bagel for a dollar. Those 800 people have lost 25 cents each.
3. The 200 people who used to pay a dollar for bagels have been replaced by 200 people who now pay 50 cents for moldy bagels. There is no gain here, unless you feel that the moldy-bagel people have more of a moral claim then fresh-bagel people. Basically, at best, there are 200 winners and 200 losers.
4. On net, we now have 800 losers.
5. With time, bagel makers leave the market, but because of the reduced profitability of being forced to sell cheap bagels -- and dealing with bagel bureaucracies -- there are fewer new people entering the bagel-making business. Even fewer bagels are made. People now find substitutes. The bagel market is wrecked. Even more restrictions are put in place to "correct" the problem. Fresh bagels become even more scarce, and are now considered a "luxury" item that only the "rich" can afford. Cinnamon-raisin bagels are almost unheard of, and usually mark the owner as being obscenely rich. 
6. Bagel makers are cast as cruel people, only interested in profits. They don't even know how to make quality bagels any more!
7. Politicians run on platforms to correct the bagel problem. More affordable bagels! (Maybe even single-payer bagels?) 
8. Really, it's the Jews behind all this. Everyone is just afraid to say it.

Public goods are those that are not produced only because the marginal benefit to a given producer is captured by others; i.e., free riders. An arguable example is pharmaceutical research, where the benefit of spending truckloads of money on research can be captured by others who then produce copycat generic drugs. The government tries to solve this problem by granting monopoly sales rights (patents) to the firm that does the research.

From a consumer's perspective, public goods are not purchased only because the marginal benefit to a given consumer is zero; i.e., they can free ride off the purchases of others. An example is the catalytic converter. The benefits of clean air can be captured if everyone else installs one; the marginal contribution of your installation is negligible and not worth the price. The government therefore forces everyone to have one in their car.

Transit

Transit producers already collect fares, demonstrating that they can capture benefits. And the utility of transit to its users is greatly increased when they use it; they don't receive the benefits if they stay at home. This is how airlines, private bus lines, cargo carriers, etc. operate. Transit systems have the same properties.

Education

Education is already being privately supplied for Pre-K and college in exchange for tuition; there is nothing about K-12 that makes it any different. And education consumers capture the benefits of education when they get a job. Also, many people see education as an end in itself which doesn't even require financial returns from a well-paying job; e.g. degrees in psychology, linguistics, etc. 

People who feel that transit and education should be publicly supplied should rely on other arguments; these services are not public goods.