[i]f you think that Chicago's teachers deserve the right to form an association to advocate, lobby, and bargain on behalf of the interests of its members (and why shouldn't they?) then you have to think that they deserve the right to advocate for ideas that may not be in the public interestWithout a rough definition of "public interest" the claim that if I support public unions then I must support advocating anti-public interest ideas is devoid of content.
Indeed, given that teachers have a set of concrete demands, both in this case and in general, it would be helpful in Yglesias offered some examples of anti-public interest advocacy or policies. Given that his example drawn from private sector unions is the increased cost associated with increased wages, he seems, although given his dunderheadedness it is hard to know, to mean that increased wages mean increased taxes.
The problem here, of course, is that only neoliberals and libertarians fully support the notion that providing adequate funding for public services is anti-public good. Paying teachers a decent wage, protecting them from the arbitrary authority of administrators in thrall to the latest educ-scam, and the like are, actually, policies that promote the public good. Smaller classrooms and more teachers make for better schools. Limiting the power of the administration or rabid maniacs riding various political, religions, or other hobbyhorses to dictate curriculum or tenure and promotion decision is another public good. And so on.
People babble on about rubber rooms and lazy teachers but the fact of the matter is that teaching is a highly competitive profession and thee most teachers care about students and want their schools to continue to improve. Assuming that they and their unions want to advocate for policies that decrease the public good is one way to assure that the best and the brightest of this and any future generation will seek to join a profession, like banking, investing, or punditry, where failure is not an option and even the dimmest of bulbs is free to fail upwards.