Answer

Not with the study of Jewish Thought, but with the field itself. There is no such field. There is what is true and what is not true, and I do not care whether it comes from Israel or from Ishmael.
Proofs for the existence of God belong to philosophy. The fact that there are Jews who are interested in this and engage in it does not turn it into Jewish Thought. Falafel, too, is not a Lod dish just because people eat it in Lod. They eat it in Tel Aviv too.
As for infinite regress, I discussed it at great length in The First Existent. My claim is that an explanation by way of an infinite chain is not an explanation, like turtles all the way down. It is an escape from explanation. And the root of the matter is that infinity is defined only potentially and not concretely, whereas an explanation based on infinite regress relates to it concretely.