I think you’re right. I didn’t think the “helper words” in the conditional should get conjugated, but I grabbed a Book of Common Prayer off the shelf and there’s a bunch of “thou shalt” + infinitive, so evidently the conditional does get conjugated (in addition to “thou didst” and “thou hast”.) Pretty sure I noticed some 2nd person weak verbs that looked like they had the same conjugation as the 3rd person (eg “Remember thou keep holy …”) I did note “he cometh”, so maybe that -eth ending is actually an older conjugation for the 3rd person that later morphed into an -s ending? Just noticed “he saith (says)”, and the confirmed -eth ending on a bunch of 3rd person congregations. Interestingly, I found a LOT of “thou shalt”, some “thou wilt”, but no “thou couldst” or “thou wouldst”. Probably because the BCP is all like, “you WILL, this is not an option, sinner.”

I don’t know though! I’m a typical English first language speaker and I’m just going with what feels right and using my understanding of grammar from my French education.

We actually do have a second person singular, “thou.” We just transitioned out of using it because ‘politeness’. Thou could useth the second person singular, but thou would soundeth quite archaic. (Think I conjugated that correctly.) You can still see it used in some religious texts in reference to God.