I think the main reason for the initial focus on calculus in intro QM is more to do with making sure you have a solid grasp of the ugly, complicated, hard way of doing things before diving into the more elegant, easy way (once armed with an appreciation of all the whacky machinery under the simplified proverbial hood). I recall my intro classical mechanics had a similar approach on a few topics, and my professor explicitly said this was the reason for it. That and the elegant way is generally more abstract, and it's important to have the foundation the abstraction was built on first.
The link you posted looks interesting and fun to read, I'm looking forward to going through it after work. Thank you!
