Which is the same as how it works in Cubase - Notepad needs to loose focus. When you “control” your browser like you describe, you are no longer in Notepad so no you cannot control your browser while in Notepad.
I also find it frustrating for doing some some things when I’ve been editing in the Notepad. But it kinda makes sense. I mostly use the spacebar to start & stop playback. If I’m in the Notepad and hit the spacebar how is Cubase supposed to know I want it to play instead of you know putting a space between two words in the Notepad. When you are inside Notepad Cubase assumes that what you are typing is text you are entering. If that’s not what you want it to do, then you need to let Cubase know by shifting focus away from Notepad.
Good news though: You can get a suitable used controller for substantually less than $450 (nanoKontrol is a great cheap option), or if you have assignable buttons on your keyboard it could be free…