It's income tax day tomorrow. The money that you were going to give to the Institute for the Propagation of the Invisible Alien Theory of Motion is instead taken by the government, who uses it to suppress that theory. For example:
Though the TA probably isn't thinking about freedom of speech, it is at issue here: can the publicly funded university - a government agent - use your tax dollars to speak in favor of one point of view and against another?
Physics TA, a government agent: Can anyone explain why this cart moves in this way?
Student: Invisible aliens are pushing it! They want it to crash -
TA, paid with your tax dollars: Excuse me; let's watch this expensive demo to explain why you're wrong.
Very well; suppose we agree that the TA - a government agent - can shut down discussion of the Theory of Invisible Alien Motion. Can he then shut down discussion of the Theory of Evolution? Or the Theory of Intelligent Design? Arkansas, among many other states, tried to ban evolution from the schools; the Supreme Court overturned the law - but only because it rested on "a particular religious doctrine." In general, they admitted "the State's undoubted right to prescribe the curriculum for its public schools." Similarly, Dover v. Kitzmiller banned Intelligent Design from the schools because it constituted an establishment of religion. Free speech is often mentioned, but not used. Even Meyer v. Nebraska struck down a law prohibiting teaching of foreign languages only because it banned them outright - the Nebraska government was free to do whatever it wanted with publicly funded schools.
Nor are these government powers limited to publicly funded schools. Any program with public funding is subject to government control. It can prevent a family planning clinic from talking about abortion; it can prevent a radio station from editorializing. Very soon, any corporation contracting with the government or receiving any government funds may be "barred from making political expenditures". (True, they could establish affiliate groups to do so with private funds, but those are "burdensome alternatives... expensive to administer and subject to extensive regulations... onerous." For example, they have to use discrete resources.) In general, "when the government appropriates public funds to establish a program, it is entitled to define the limits of that program."
At least to some extent, this principle is vital. We have a Statue of Liberty; must we also have a Statue of Autocracy? We have a National Endowment for Democracy; must we also have a National Endowment for Fascism? Even if we agree to end the National Endowment for Democracy, tear down the Statue of Liberty along with Pleasant Grove's Ten Commandments monument, and rename Washington, District of Columbia, lest supporters of Benedict Arnold and Lief Erikson feel slighted, we are left with a Constitution which compels Federal officials to support a republic above a dictatorship and honesty above perjury. "The law knows no heresy, and is committed to the support of no dogma, the establishment of no sect" - but the government, if it is to do anything at all, must be!
Fortunately, there are some checks which would prevent this from sending a pale of government control over all our liberties. The Supreme Court has also ruled that a legitimate government purpose "cannot be pursued by means that broadly stifle fundamental personal liberties when the end can be more narrowly achieved." On this basis, they invalidated laws requiring teachers to list organizations they'd joined or refrain from "'treasonable or seditious' utterances or acts". Public employees may campaign off-the-job, and corporations with government contracts may set up independent organizations to campaign.
Whatever potential these restraints may hold, however, they might not be enough. Even if the government is not allowed to "broadly stifle" liberty, it can still discourage it. For example, the courts recently allowed the University of California to not count high school courses taught with creationist textbooks, in language which would allow them to reject whatever other high school courses they wanted. Similarly, even though a privately funded program is quite free to talk about political editorials, abortion, or anything else, many customers of the federally funded competitor will look no farther and remain ignorant of the other options. If the federal government raises taxes to provide for healthcare, counseling services, or any other program, people will have less money to fund those programs privately. Many people have argued that newspapers and radio shows must give each view ample airtime to avoid drowning anyone out. If a private corporation can risk drowning other voices out by speech, how much more can the government do so by speech and taxes?
However, despite all these problems, the government must have the right to define the scope of the programs it funds. So, the only possible way to avoid drowning out free speech is to severely limit the number or scope of government-funded programs.