From what I'm understanding, the relevant part of the treaty was that the land Mexico was ceding to the US government came with a proviso stating that individuals who had been granted portions of that land by Mexico prior to the end of the war be allowed to retain ownership of it.
However, the effects of the treaty only extend to the exchange of land between Mexico and the US at the end of the war. Whatever changes to land ownership that occured after that fact supercede it, as do any changes that occured to the laws governing that land.
It just so happens that both the public trust doctrine in the California Constitution and the California Coastal Access laws were both enacted after the treaty as, and do in fact supercede it.
Jose Antonio Alviso may have retained private ownership of the land, but it was still (by virtue of the very treaty being cited) territory of the United States of America and subject to the laws and jursidiction thereof.
When that territory was incorporated into the State of California, the Constitution of said state came into effect and the land became suject to the laws set down therin, including the provision establishing public beach access.
Then in 1976, when the California Coastal Act was signed into law, the state of California empowered the California Coastal Commission with the means and authority to guarantee and enforce that public beach access.
Despite being privately owned, the land in question is still subject to these laws. Public beach access is legally required of the land holder, and the California Coastal Commission is legally empowered to enforce it.