Sounds nice if you can do it, but it also seems to require a deep familiarity with the author that is impossible, short of actually knowing them personally? This is what makes me nervous about anything "Straussian", in most cases it requires a lot of discipline to avoid just filling in gaps with what you yourself wish the author was saying.
In the Melzer Philosophy Between the Lines book mentioned in the article, the author does a pretty throrough job of making the case that pre-modern philosophical writing defaulted to esoteric style, while in the modern era we default to exoteric.
He does this by pointing to the allusions and references they use and the clear contradictions within them - often inversions of meaning from the original sources that are alluded to. Socrates makes these errors, as does Machiavelli. If you assume the errors are mistakes, the work's apparent conclusions become more shaky, but if they are intentional puzzles, the work gains additional meaning. We often use puzzles as teaching devices and writing isn't really different in that respect.
It's not really different from reading for literary subtext, it just applies that toolbox more widely. I believe it's more useful for writing than for reading because esotericism makes it easy to encode something legible only to intended recipients. I would not expect to parse new wisdom from the utterings of a CEO, as in article's example, until I become familiar with corporate culture in general.
I'm curious if Melzer addresses Discourses on Livy or only The Prince. In an introductory political science course at my college's Straussian department, The Prince was read early as an example of esotericism, and that reading was (implicitly) proof that we should read other texts esoterically.
My understanding is that Discourses on Livy is far less esoteric, though. It doesn't surprise me that particular works are esoteric, but does Melzer successfully advance an argument that entire bodies of work or even an entire era should be approached that way, without needing to evaluate each text on its own?
Beware, Straussian reading largely means reading a text and making it mean anything you want it to. Your evidence is the “esoteric meaning” you uncovered. The magic of this is when someone tries to debate you using the same source you can retort that your critics just haven’t read close enough or have not grappled with the material the same way you have. Despite being advertised as otherwise it is a rather unscientific method of reading a text and perhaps for this reason it is so prolific among the American right.