If you want to take this stuff seriously, absolutely study computer science, and a little computer engineering as well. When you really understand the fundamentals, you can pick up the rest.
If you want to get hired as programmer in AAA game development, then your job role starts to become more specialized as you move up, but that's true in a lot of environments. And jobs for AAA game development are admittedly competitive, so the more value your bring, the more likely you are to be hired. Sometimes that means being an uber elite coding ninja, but it also might mean that you are able to wear a lot of different hats, including skills that aren't programming.
Anyway, I just want to say that jumping "in with minimal programming experience and try to muddle their way through by watching tutorial videos and stuff" is awesome, actually, especially if your end goal is making games itself. Just do it!
Balatro, a recently-released solo-dev indie game, is a great example of this. I saw two tweets the other day that, together, support your point: "Balatro has a 5000 line if-else chain"[0] and "in the 3 months since Balatro has launched, it has been collectively played for a total of 6200 years"[1].
[0]: https://www.reddit.com/r/gamedev/comments/1cbcmr0/how_is_it_... (couldn't find the twitter link directly but summarized it from this reddit post)