It's at least as private as third party cookies. And unlike third party cookies there is a path to improve privacy as technology improves.
Some of the required technologies (private model training, debuggable trusted execution environments) are still research topics, so some sacrifices have to be made until it can be deployed.
Sorry, I'm not a WSJ subscriber. It wouldn't surprise me if Google are being squeezed between two organizations with different goals though.
Ultimately the Privacy Sandbox has dozens of different proposals, and each is on a separate standards track. It's not a singular technology.
I will say that many of the proposals do directly improve user privacy, or offer more-private alternatives to existing APIs. But I'd also be surprised if there weren't objections as well. It's the web, and scrutiny is important.