Now, you can say "That's not magic, it's science/psychiatry pretending to be magic", but if that's the case ; what was magic supposed to be before D&D invented the concept of the magic missile?
I will say it, it is just psychology
Magic, by definition, invokes the supernatural. Since, by definition, nothing supernatural exists... Magic is and has always been useful in storytelling or fantasy. Without fiction life would be pretty boring. From a practical standpoint magic is also useful for thought experiments or tricking your children into behaving around Christmas.
It depends a lot on your defintion of what we mean when we say "magic" or "science".
I think part of the contention between yourself and Jeoshua in this stems from differing definitions of "Magic" and "Science", since you seem to view those two things as a thought process, whereas Jeoshua is thinking more in terms of whether or not something
appears to be magic.
In the ancient world, the Sun was magic - The Egyptians believed that the world was flat, and bounded by very tall mountains, and that the Sun was the boat their god rode through the sky, and during the hours of the night, descended into the underworld to do battle with the forces of Apep, where if the Sun were ever destroyed, Apep would be free to swallow the whole world of the living.
That's all magic, replaced by our more modern understanding of things like gravity.
There's also the sort of magic that takes place in modern fiction. Consider Mass Effect, where there is a fictional substance that allows heavy objects to become weightless, faster-than-light travel, the ability for people to levitate objects, or make force fields or use powers of their mind to rip physical objects apart.
This is all given a science fiction veneer, but it's functionally the exact same
effect as magic.
Even worse is something like Xenosaga, where the characters have "nanomachines" that are just obvious stand-ins for magic in a supposedly technological setting, and even go so far as to have The Four Greek Classical Elements "castable" in nanomachine form, along with standard healing spells.
Then you get things like Final Fantasy 7, where "magic" is basically just gasoline that glows - you can run a truck by pouring magic into the internal combustion engine, and the giant steampunk factories run on drilling an exhaustable supply of "magic" from the ground, where society will collapse if all the "magic" is drilled out. The good guys want to switch over to technology that doesn't rely upon drilling out the last of the "magic" from the ground before it's too late, and live in peaceful, eco-friendly sustainable villages. (Nope, no parallels to fossil fuels, here.)
It's just technology which people have applied a fictional power source to it.
Regardless of how the actual alchemical system is set up, as long as "Component A + Component B ==catalyst=> Product" is true, you're basically talking about "science" in the sense that it is a predictable, explainable reaction, (the way that you have been talking about it, and the way that "magic" exists in Final Fantasy 7). We're just using eyeball grass or mushroom wood that is always 0 degrees Celsius as a component, and the rest is just a semantic argument.