Obviously this is not a super important thing in the scope of our world and my day job. But still, it feeds my curiosity (as does this site) and that alone is more than worth the investment to me.
Also, being able to design and manufacture a plastic part in a matter of hours any time of day or night has become a really useful tool. Only a few days ago the window latch on the front door of our apartment broke off and we've already had too many break-ins. It was too late in the day to go buy a new one but the printed one works really nicely.
Sorry for nit-picking.
I like your reasoning tho; I got mine when it was on sale, and just before the pandemic - so I have plenty of learnings about PLA, ABS, PETG, temperatures, glue or not, surface, and most of all -- plenty of piles of material creating 'modern art' rather than my planned masterpieces. '-)
Perhaps, distilled, it's this: having a 3D printer, by itself, won't turn you into a MechE. But, if you do have a 3D printer, you can learn the basics of ME and try out some practical ideas with real parts.
I specifically got my 3D printer to make gears. I do have a Sherline mill, as well as a 12" (Chinese) lathe; so I do have the specific tools to create involute gears out of metals. However, since I planned on doing so many different gears, as I wasn't really sure what I needed, AND the fact that with subtractive manufacturing, if you make certain mistakes, you throw the whole piece away and start over -- 3D printing makes creating new parts as trivial as starting it up again. The other total win of 3D is e.g. Thingverse where you can d/l a design -- it's the closest thing we have to Star Trek's replicator.
Lastly, Michael Faraday wrote an entire book on candles. https://en.wikipedia.org/wiki/The_Chemical_History_of_a_Cand... So much can be learned from little!
Mechanical engineering consists of learning tools for analyzing designs, such as fluid mechanics, materials, solid mechanics, thermodynamics, statics and dynamics, control system theory, etc.
I think you can really pick up Fluids, vibrations, and advanced mechanics on your own with that background, if you end up needing it. But to the larger question, yeah you need calc 1-3, ODEs, and actually a good bit linear algebra to be able to understand and apply 300-level engineering and beyond.
Becoming a wizz at industrial design was never my goal. There's universities for that. But it's wonderful to see how everything comes together.
Also, imagining something only in your head and then seeing it come to life on the screen and then in the real world is a magical thing.
I guess carpenters and sculpters etc get this all the time but I totally lack the manual skills needed for that. The 3D printer solves my handicap of having two left hands.
Most people that buy a 3d printer download some models from thingiverse and then get bored with it. It's when you learn design that it really becomes amazing. And it's not all that difficult at all.
On the other hand, no better way to learn an STM32 and automatization than looking into Prusa Mini firmware github ;-)