Generally speaking, most of your knowledge will be ascertained through initiative. College is great, and in my humble op-onion (though others have had different results, and may have a different perspective) a generic software engineering degree has the potential to open up more doors than a Game Development degree. So I would recommend sticking it through if you're having second thoughts.
Insofar as the engine v.s. custom engine it's probably worth noting that the term, "Engine" gets thrown a bit much. Many people extrapolate it to many things. Generally, when i think engine, I think the works, scene-graph, full blown editor, drag&drop shaders, nice high-level api for the object logic, etc. While this is a route you could go. It could take awhile. That doesn't mean it's not worth it, and you wouldn't learn a lot through the process it's just a matter of how quickly you want to get items into that portfolio.
To clarify,
You can still create a game without any engine. Right, kinda what you eluded to in your post. This takes us down a step further with actually _talking_ to an API that'll assist you in rendering graphics, and in most cases, only that. (Take Direct3D for example. That api has been trimmed down in recent years to basically be an interface for just your GPU pipeline. While optional APIs like xinput, and friends existing under the umbrella of the DirectX platform). I can code a game with the API boilerplate, and game logic tailored just for that game. Sure, this will likely, if coded well, result in re-useable code, but that's a far cry from an engine type experience.
There are also lots of game oriented frameworks out there. Tools that don't truly rise to the level of an, "Engine experience" but abstract lots of the underpinnings you would need to handle yourself at the API layer. SDL is the first to come to mind, but I'm fairly sure that's 2D exclusively. However, there are 3D rendering frameworks out there as well that aren't coming to mind atm. I'm sure other members could fill in that omission
Working with pre-built engines is also fine! You can do some great stuff very portfolio worthy with most engines nowadays. From what I hear from the grindstone around the forums is UE tends to have a slightly higher learning curve than unity, but I'd still think both would be a good choice.
In summation, all the options you mentioned are fine, it's just a matter of what you want your portfolio to look like, and how quickly:
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Do you want to show off a WIP engine, that, sure, doesn't really render much of anything interesting to the screen, but has a solid design.
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A game that is of limited means in the art department, but has some cool gameplay mechanics
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And (optionally) some engine experience Unity/UE4 thrown in