I am of the personal opinion that until complex numbers and manipulation of infinite series is introduced that it is pointless to introduce hyperbolic trig functions as I had learned them in that order and fared just fine.
That being said the rest of the math worlds seems to disagree insisting on learning hyperbolic-trig indentities and integrals before showing most students what these things actually are.
The reason they aren't seen too much though and given the 'fair treatment' is because it's not pedagogically useful to focus too much on them though since what ends up happening is that students need to be taught a plethora of trigonometric-style tricks without much (if any) intuition or proof when they are first introduced.
By studying Trig first, followed by taylor series, and complex exponentials one much more naturally arrives at the theory of hyperbolic-trig and has an algebraic means of understanding exactly why they behave the way they do.
For people that insist on geometric intuition: the diagram below:
With the use of some heavy algebra (and knowledge of tools such as arc-length formulas) can help them derive all the trigonometric-style tricks
I would think this is a useful method too to learn them but have never seen this done, EVER, in a class.
Personally, I think the best order of learning is: differential and integral calculus for simple algebraic functions followed by taylor expansions, then followed by an introduction to trigonometry whereas as soon as some of the basics have been learned, students are more than well equipped to derive taylor expansions for the trigonometric functions, determine their complex exponential formulas, etc, etc...
Lastly hyperbolic trig, if anything should be something students derive on their own attempting to solve their associated differential equations.