Alright, after some testing and waiting for packages to arrive, I've finally found out the problem.
So to start, I've got myself a multimeter and checked if 3.3v and 5v were actually present on the breadboard, which they were using the cobbler. Afterwards, I've removed the cobbler completely and used 3 jumper wires and a resistor to check if it was reading low when the GPIO was connected to ground and if it was reading high when connected to 3v3, as joan suggested. It did.
As everything was looking fine pretty far, I've decided to add the button and see if the GPIO would measure low when the button was not pressed, and high when it was, as pointed out by NomadMaker.
So using the same script, it did measure low when not pressed, but the state did not change when the button was pressed. (It's not possible to rotate the button 90 degrees as it only goes in the correct way).
At that point, I was pretty confused. These momentary switches had come from a kit with the breadboard and the pi. In the meantime, I had ordered some bigger buttons to test them out as well, as Shreyas Murali suggested that they may not be making good contact with the breadboard, due to the short leads.
When they've arrived, I've switched (pun not intended) these buttons with the bigger buttons, and right off the bat, I've noticed that the new buttons were way larger and had significantly longer leads than the ones that came with the kit.
So, after replacing them, booting up the pi, and running the original script, which would print out the "Button pressed" message whenever the button was pressed, or when the state of the GPIO was high, the result was that... The message was printed. The message was being printed whenever I was pressing the button, as the GPIO state was changing. That's right, the previous buttons were not properly making contact with the breadboard. The momentary switches that came with the kit, were not making contact. It was that simple. What a waste of time! And on top of that, the switches on the cover of the kit are similar to the ones that I had ordered and completely different from the ones you get in the box. Thanks to everyone that tried to help!
The buttons were not making good contact with the breadboard. Anyone having this exact problem with the same switches or the MCM Electronics Project kit, buy your own switches.