Every book I've looked on the $t$-distribution rushes through the rationale behind it. They don't even cover the thinking that gave rise to it, let alone give a rigorous derivation.
I'm looking for a book that does not short-change these questions.
(By the way, the kind of treatment I'm hoping to find would be far more extensive than what can reasonably be expected from a typical MSE answer. This is why this question is a reference request.)