Public reason liberals' self-conception differs from how others view them.
Public reason liberals view themselves as inclusive theorists try to construct some sort of great compromise. People disagree about the good and the just. In order to promote peace, stability, and cooperation on fair terms all people inclined to cooperation can accept, we should...[insert their theory here].
But, as I've been arguing for the past few days, to everyone else this simply comes across as yet another parochial theory, but without the arguments. Everyone acknowledges that we disagree and that we should work together anyway, and every theorist has their own view about how to resolve this problem. Further, every theorist can say that their view accommodates some views and not others. Liberalism is a big tent--but so are other theories. Further, it's not antecedently given that being a big tent is good. Perhaps we are directed by God to instantiate the City of God on the hill and tolerate nothing else. If that's the correct theory, then public reason liberalism is not merely false but straight up evil.
Here's further evidence that it's just another parochial theory: The public reason liberals disagree with each other about what public reason entails. So far, no surprise--that's true of every big political theory. But notice the shape of their disagreement. People who were classical liberal or libertarian before they go into public reason liberalism always conclude...wait for it...that pretty much only classical liberal politics can pass public justification. People who were left liberal, such as Rawls, conclude...you might want to to sit down for this...that left-liberal politics is publicly justifiable and other stuff isn't. Socialist liberals conclude...you may notice a pattern here...that socialist liberal politics is publicly justifiable and nothing else is.
What people do is take the basic concepts of the theory--such as reasonableness, justification, publicity, rationality, cooperation, coercion, and do on--and custom-tailor them to ensure that the final theory comes out defending and requiring what they antecedently believed. Rawls's 1996 Political Liberalism ends up arguing that a barely modified version of his 1971 A Theory of Justice would be the overlapping consensus of all reasonable people. When, say, Gaus or Vallier do the public reason project, they conclude instead that it has a classical liberal tilt.
The public reason project is a bit like American constitutional law. Most lawyers and legal theorists accept that the Constitution is authoritative, but they also believe their own ideology is correct. So, to avoid conflict, they reverse engineer a theory of constitutional interpretation and constitutional law which makes it so that the Constitution fortuitously happens to allow what they want it to allow and forbid what they want it it forbid. They can't always make it work, but 90% of the time, they can come up with an account of why it turns out the Constitution says what they themselves thought justice required before they studied con law.
Something similar happens in Biblical interpretation. Christians believe the Bible is authoritative, and they also have their own private views of the good and the just. So, they reverse engineer a theory of interpretation which fortuitously makes it come out that God wants what they themselves want. If you are inclined to hate gay people, you take Leviticus very seriously. If you are inclined not to do so, you argue that Christ released us from various ceremonial restrictions on behavior or that this passage of Leviticus has been mistranslated or altered over time.
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When I think of people who take public justification seriously, I think not of Rawls or Gaus, but Peter Singer. Singer writes for both a popular and elite audience. He doesn't beg the question in favor of his own theory or start by assuming his theory is correct. He begins with widely shared premises pretty much everyone accepts and then attempts to show how they lead to surprising conclusions.