Well, the thing is, you don't need 300 fonts to do something nice. And the issue you mention, all designers face it!
The reason is: when you have 3 paid quality fonts, you wish you had 30. And when you have 50, you wish you have those 10 new ones that look so amazing... ;) There is indeed a big part of investment in graphic design and that goes for stock images, fonts, equipment, office supplies, Pantone book, etc. And there's always something you wish you had. I think most designers are like little squirrels, they gather resources and tools as they work, and over time. You need to start somewhere.
If you don't have a budget for fonts, you can start with the free ones and tweak them, adjust the bad kerning, etc. They're not all perfect but let's be honest, that in itself offers you a great way to practice and develop a keen eye with typography! Sometimes you can use them for your titles and combine them with system fonts for the main body. It's less work and the system fonts are in fact high quality.
There's also very nice fonts that are not that expensive to purchase and that you can use on many projects. When you purchase fonts, look at it as a long term investment and try to stay away from trendy stuff if the investment is big. Some fonts only cost $25-30 for a whole family. That's worth it... that's barely the cost of 5 lattés coffees or 2 meals at a restaurant.
There's open source fonts that allow you to work on any kind of projects. Nice selection there too.
If you have an Adobe subscription, there's the TypeKit fonts that have some nice fonts (but yes, still limited) too.
I wouldn't recommend getting into pirated stuff.
As for using system fonts, I personally love using Georgia (eeek!) and Helvetica but you'll never see any of my design done with Papyrus or Corsiva or the typical system fonts that do look... amateur. There's nice classical system fonts that you can use with other free ones or low cost ones. As a student, you won't get judge by how great your fonts look but usually how clever is your font matching... or how you used something ugly and made something great out of it! I do think at least employers in design firms can recognize this and are aware of the reality of being a student with limited budget.
TLDR: Don't judge fonts by the price or "premium" status.
Where do professional designers "go" to look for typefaces?