I don't think we disagree as much as he thinks we do. As he says, I agree that the Fifth Amendment bars private-use takings as a result of the original settled meaning of "public use." The question is whether anything happened in the nineteenth century to unsettle that meaning. What I found especially interesting about his paper is that courts apparently did begin to allow private-use takings (premised on indirect public benefit) in the early- to mid-nineteenth century -- much earlier than I had thought. My question (and I only meant to raise a question) is whether that is enough to cast doubt on the meaning of the Fourteenth Amendment as applied to takings.
After reading his post, I'm more clear that Professor Somin takes the view that the privileges or immunities clause directly adopts the textual right expressed in the Ffith Amendment (as it was understood in 1867-68). If that's right, then I think he is on strong ground in saying that the adoption of a different meaning by a minority of courts probably isn't enough to unsettle it (although I'd like to know more about the circumstances of those decisions and how they were received and defended). My main point, though, is that that's not the only way to understand the privileges or immunities clause.
He has the following especially interesting paragraph in response:
Moreover, it is strange to interpret incorporation as only including those rights that were “deeply rooted” in the practice of all or the vast majority of state governments. The whole purpose of incorporation (and of the Fourteenth Amendment more generally) was to force states to change some of their deeply rooted practices. For example, the incorporation of the First Amendment’s Free Speech Clause was intended to force southern state governments to stop their longstanding policies of censoring abolitionist speech and speech advocating equal rights for African-Americans. Under Ramsey’s approach, the incorporation of the First Amendment would not include the a right to engage in abolitionist speech, because a significant minority of states had a longstanding practice of repressing such speech. Similarly, part of the purpose of incorporating the Takings Clause was to prevent states from abusing the property rights of African-Americans and southern whites who had supported the Union during the Civil War. Allowing states to take property for whatever reasons they want was incompatible with that objective.
To be clear, the "deeply rooted" approach is not my approach (in the sense that I think it's correct): I only think it is a plausible view of privileges or immunities that should be considered. But he makes some good points against it, and raises the important question of the extent to which the clause was supposed to change state practices. In this regard, his point connects with an argument Chris Green made in his paper at the conference (which I'll mention in a separate post) that perhaps the Amendment could be read to adopt the views of the Northern states on what rights were "privileges or immunties."