You can, and you can use that cake as an ingredient in something else! But if your goal is to "make a cake" and put your own touches on it, likely that weighing out the ingredients is not worth your time outside of the educational context you describe wrt teaching kids. (e.g. "here's how cake is fundamentally made, and later here's a box mix that takes care of the most boring parts and works better than anything we can make at home without substantially more effort").

This strikes me as the same sort of semantic game one plays when claiming that "humans are actually fish"; while technically true, this drains the term of its utility.

While I take your point about economy, in my kitchen it would actually take less time to make a cake from flour, sugar, butter, and eggs, because we always have them - whereas buying a Betty Crocker mix would require a trip to a grocery store, and not the convenient one around the corner I usually visit.

Nor are the results entirely fungible: I'd always prefer to avoid ingesting preservatives, artificial flavors, corn syrup, etc.

What? If you want to make your own touch then varying the ingredients is one of the most impactful things you could do.