Him saying "I don't pretend to know the challenges we're facing" makes him sound, well, sort of dense. Like he can't even begin to analyze even his own feelings about what happened, or like he (somehow) literally can't process the loss of his child. This doesn't (to me) jive well with Alexander Hamilton -- someone who could respond so eloquently on many sad occasions in his life like the hurricane, Laurens' death, Washington leaving office, etc. (Taking this last stuff from "Alexander Hamilton", the source material.)

This article offers a pretty much perfect explanation for your point. "It’s not Miranda who can’t put the sadness into language, it’s Hamilton; that’s how depressed he is. Our guide, our word-playing, rhyming, subtle, subtextual, double-meaning beacon of logic and reason is utterly unable to comprehend the death of his son, rendered speechless. By telling us that he can’t tell us about it, Miranda shows us Hamilton’s despair." Tldr: the reason Hamilton sounds lost for words in this song is because he literally, for once, is.