You are, of course, correct. I am surprised by how many people who would otherwise identify themselves as Christians apparently don't understand the Messianic typology of the Passover lamb. This is a paint-by-numbers stuff -- what do they think John the Baptist meant? (i.e., John 1:29) -- even if (attempted) Kasich's expression of it was rather poorly done in this instance.
And, as you say, the setting in which he related it was, on its face, rather "provocative." If Kasich actually intended it as such, I'd have to give him credit (although I don't know how he thought do so would help him politically); if he was being inadvertently impolitic, well, then, Kasich was, in his own way, as off-base as Trump's referring to "Second Corinthians" as "Two Corinthians" during his speech a few months back at Liberty University.
I don’t think he was proselytizing. I think he was trying to build a bridge and show commonalities. I read the article, and no one seemed out of sorts if I understood the author correctly. I never gave Trump grief over the 2 Corinthians comment, because that’s how I refer to it more often than not when I’m preaching. I might say “turn in your Bible to Second Corinthians, chapter 1 and verse 1” when opening, but often when citing, I’ll just say, “Like it says in 2 Co 1:1”. Sometimes, I’ll even say “Co”.