@PeterShor Thank you so very much!

Sorry for the late comment, but is there any chance you could point me to some references for the definition missing the uniformity condition and the subsequent paper published by the three mathematicians?

Here's another data point. Hausdorff began his classic textbook Grundzüge der Mengenlehre (1914) with the claim "Die Mengenlehre ist das Fundament der gesamten Mathematik." on page 1. "Set theory is the foundation for the whole of mathematics".

Indeed. And in fact in the argument above we could have picked any $z$ to begin with. For instance a $z$ that codes the ordertype of $Ord^M$. The same argument will supply an example of two generic extensions with no common extension with the same ordinals, which the blockchain argument in Joel's paper and answer above can also demonstrated.