I have had a weak shift in opinion towards the value of attempting to quantify and utilise weak arguments in internal epistemology, after our in person conversation and the clarification of what you meant.

Ok. See also my discussion post giving clarifications.

I have had a much lesser shift in opinion of the value of weak arguments in rhetoric, or other discourse where I cannot assume that my interlocutor is entirely rational and truth-seeking.

I think that the most productive careful analysis of the validity of a claim occurs in writing, with people who one believes to be arguing in good faith.

In person, you highlighted the problem of the first person to give arguments having an argumentative advantage due to priming effects. I think this is much less of a problem in writing, where one has time to think and formulate responses.

I have not had a substantial shift in opinion about the history of mathematics (see below).

My view on this point is very much contingent on what Euler actually did as opposed to a general argument of the type "heuristics can be used to reach true conclusions, and so we can have high confidence in something that's supported by heuristics."

Beyond using a rough heuristic to generate the identity, Euler numerically checked whether the coefficients agreed (testing highly nontrivial identities that had previously been unknown) and found them to agree with high precision, and verified that specializing the identity recovered known results.

If you don't find his evidence convincing, then as you say, we have to agree to disagree because we can't fully externalize our intuitions