I, armed with the following from the OED, dispute the assertion that exegesis means Bible learning. OED tells me the word means
An explanation or interpretation of a text, esp. of scripture or a
scriptural passage. Also more generally: a critical discourse or
commentary.
so the word is especially but not exclusively applied to the study of scripture.
(I also dispute that many outside the church are unfamiliar with the word.)
So exegesis is available to be used when making a critical reading of a text, religious or otherwise. And in this degraded age, when so few are concerned with the study of scriptures, I don't see why we secularists oughtn't to say
Hang on a minute, that's a useful word, we'll have that back from the
believers. Why, I might call my next PhD thesis The Exegesis of Elon
Musk's Twitterings.
And, while I'm in the mood for an argument, I'll take a wee swipe at originalism and textualism as replacements for exegesis. They are, rather, approaches to exegesis.