As someone who currently works for a major studio, I’d like to offer a few pieces of advice…
For starters, I wouldn’t necessarily think of taking the courses in terms of “to launch a career in game dev” or “supplemental adding of knowledge” as the two ideas are not mutually exclusive. Instead, I’d look at knowledge gained via coursework as a foundation from which you can build upon in pursuit of landing a job in game dev. One thing game companies like to see when hiring junior developers is any sort of project work you’ve completed on your own, out of personal interest. This shows initiative, and also demonstrates that you have the sustained motivation to work on games over the long term.
Another thing you should ask yourself is what role you want to pursue in game development. In general, the disciplines are separated into art, design, engineering, production, and testing. It’s good to decide and focus on the area you see yourself pursuing as a career. There are also specific roles that act as a ‘bridge’ between certain disciplines such as Tech Artist or Test Engineer for example. In fact, both of the roles I just mentioned are usually in high demand, so you may want to do a little research on them and see if either appeals to you.
On the topic of Unreal and Unity - I’ve used both, but I am a bit biased because I have far more experience with Unreal than I do with Unity. My personal opinion is that when you are just starting out, choose whichever tools you enjoy working with the most. Perhaps you really enjoy using Unreal or Unity, or you might be interested in an open source engine like Irrlicht or OGRE 3d, or perhaps you want to go even lower level and build a game on top of SDL. At any rate, the point is to explore the amazing tools that are out there and consider the availability and quality of learning resources available.
Now for the part where I have to include a DISCLAIMER. I am by no means advocating against learning engines other than Unreal Engine, but I do want to say this: IF you are really serious about working at a major studio, having experience with UE is generally going to give you a competitive advantage in the application process. Of course, you need to consider the studio and the type of project you are working on. Some studios develop great games with Unity (especially for mobile markets). However, UE experience is highly transferable, meaning that several studios (including the ones I’ve worked at as well as several of my colleagues) use UE as their dev platform. You may even be more likely to be considered by studios using their own proprietary engine since C++ is still the industry standard (if engineering is your goal).
I could go into much more detail but there are already many great resources out there that discuss landing a job in the game industry, including a dedicated course right here on GameDev.tv. My goal was just to give a quick summary of some things for you to consider. Good luck in your journey! 