nah i like removing noise. for me it's more like trimming the fat off of a chicken, to extend the cooking analogy.
how basic is your noise removal process, exactly? a gate is a gate is a gate, but you can also do wacky stuff like sidechained phase cancellation, or even wackier stuff like this process that evilwezil devised.
first thing i would check is that whatever you're recording with isn't adding extra noise (all hardware will to some extent but it should be negligible).
the high pitched gameboy whine is around 9.25KHz, so run a parametric EQ/notch filter over that frequency with a high Q and it should go away without affecting the rest of your sound too drastically.
edit: oh, but if you're using one of those specific "noise removal" effects then yeah maybe don't do that and instead just use a gate and EQ