Yes, it affects the outcome in an absolutely critical way, but potentially in a much simpler way than you might think. You need to be able to get it to sound right at the source, whether it be through an amp or through software. You can't generally turn it into something that it is not after the fact with any amount of processing, and as you have noticed, the more you process, the further you move away from the natural tendency of the instrument.
The actual guitar/amp/whatever is constrained in what it is able to output, but it is purely up to you to determine if that is what you want or that is what you don't want. You can't turn it into something else.
Ultimately, if you can record an instrument in such a way that you don't need to even touch an EQ, or you have to do very subtle changes to it to help it work in the mix, that will make your life easier, more fun and will give you better results.
Being able to hear what is really coming back at you accurately, recognizing what it is you are hearing and understanding how to work it is basically what it is all about if you want to be able to do it yourself. I don't claim to be there myself, but I can tell you that my former monitoring environment was holding me back because it lied. I could produce a mix that sounded absolutely stunning to me when I was sitting in the sweet spot of a very bad environment with very inaccurate speakers. That mix sounds absolutely terrible everywhere else.
So monitoring does matter (including the room) because if it is wrong, you can still develop personally and create results that sound good in that bad environment, but that will not translate across other playback systems.
You want your art to be presented in a certain way. How you accomplish that, whether it be developing and investing in a recording and mixing endeavour or hiring it out, depends on what you are in it for.