I liked the subtitle: "a system for doing mathematics by computer", as well. I don't know why it is not used anymore. Perhaps it was a marketing decision, since most people have a much more limited view of what maths is than what it really covers.

I regard the term "computational essay" as emphasizing the essay part. Although is is slowly changing, it was the case that you could get a degree in maths without writing anything other than cryptic proofs, if that. The point of the initiative is that you can write an essay exploring ideas that happen to involve computation as part of the exposition. What Mathematica (Wolfram Language) lets you do is to deal with ideas rather than symbol manipulation or bit twiddling.

I think that this is the aim of Conrad Wolfram's Computer-based mathematics. In most presentations, he (and a lot of others) go through the four steps in solving a problem using maths. The computational part is just one step. The (computational) essay is one means of integrating all four steps into one document.

In this forum (and stack exchange) as well as a lot of the educational materials from Wolfram Research, there is an emphasis on the nuts and bolts of using the tool, rather than placing the tool in context. One exception is Stephen's live-coding exercises, where he explores an idea using the computer, the way that Feynman might use a yellow pad and #2 pencil.

This idea needs to be greatly expanded, of course, and this method of presenting the function of thinking using a computer needs to be embraced by people from other disciplines. I studied music composition in my youth, and some of my teachers would take this approach in showing how to take a musical idea and explore what can be done with it. Straight lectures have their place, but they are often too polished so that a student has no idea why the lecturer is doing something or where the ideas came from. This can be mystifying to non-majors, and can end up with maths majors (physics/chemistry or engineering majors) treating maths as a cookbook for solving problems rather than as a creative act.

As a concrete example of the problems this causes, I developed a real-time process control system for a large clinical lab. It was a big success, and in order to make it work, I had to develop some new maths. my boss, who had a Ph. D. in analytical chemistry, could not accept any mathematical techniques that he had not learned in his statistics course from the 1970s. For him, maths was a frozen body of knowledge.

As for getting Mathematica in the hands of everyone, I agree in principle. Mathematica was bundled with NeXT computers at one time, and it would be great to bundle it (with a one-year "home use") license with every computer -- or iPad. I have mixed feelings about open-source. I think that there is evidence that open source projects simply cannot bring the level of polish to the software or the level of technical support that is required for a commercial product. Open source coding libraries, etc. are generally written by experts for experts, and there is an understanding that the user knows how to fix things. Think of the difference between Linux and OS X -- both of which are based on UNIX. Back in the dark days for Apple, I looked into switching to Linux when it looked like there was no future for Apple. I found that I could make it work, but I had already been coding for 20 years at that point.

The real issue is not open-source or cost, but acceptance. The cost of a student or home license for Mathematica is less than many people pay for Netflix or Amazon Prime. I realize that there is a large percentage of the population who cannot afford this, but that is a sociological problem, a problem with a solution. The real problem is that a large percentage of the population simply has no idea what maths is, and there is a social prejudice to simply hate it. This problem has been hashed out in the popular press and by people far more expert than I in this field.

For this to change, the way maths is taught in K-6 needs to change. There are encouraging signs, but I think that a lot of the success stories are limited, and the vast majority of students are still taught the 'traditional' way. There is evidence that it will be necessary to change the way pre-schoolers encounter maths, but that is a real sociological problem, and with the current anti-intellectual mindset in the US, it is not likely to change. In these areas, Mathematica (any computer algebra system) is of limited use. From what I have read, hands-on learning is more important than computer proficiency (in any field, not just maths and science), and effective learning can take place without a computer until age 10 or so. What is far more important is to have a teacher who loves maths, and there are a lot of otherwise effective K-6 teachers who are math-phobic. (It may have been a while ago, but most of my elementary school teachers were in this category.) Better education of teachers is one solution, but more important, teaching as a profession needs to be respected. (Better pay is just part of the solution.) We know that this is possible based on the experiences from other countries. [Sorry to make this part US-centric, but at least some of my observations have more generality.]

As for the rest, learning the specific syntax of Wolfram language is no different than learning the menus and conventions for a word processor or spreadsheet. I think that Wolfram Language has an easier entry than most other technical software, particularly if you use natural language input as an on-ramp. Having easy access to real-world data makes applying maths to a student's experience much easier -- no more trains colliding problems or made up nonsense. Being able to easily change things (as in Manipulate[]) makes it much easier to get a grasp of fundamental concepts than to work through a bunch of very similar exercises.

I will be putting these ideas to the test. Over the years, I have developed a large number of gaps in my mathematical knowledge, and I intend to fill them by using Mathematica. I will be making lots of (computational) essays, demonstrations, and explorations as a matter of course. Whether any of this work leaves the nest is unknown at this point, but my goal is insight, not publication. We shall see.