When applying for a Master's, I know that extra-curriculars and part-time work barely count for anything in your application. However, say I have an average GPA but my CV shows that I was busy with several other non-academic activities during my degree, wouldn't that show that my GPA is not an indication of my full academic potential? I mean if I'm able to maintain an average GPA in a rigorous STEM course while working part-time, playing in a sports team and participating in many other things on and off campus, surely I will perform better if I just focus on my course?
Your GPA is already just one metric that may be considered in an admissions process. If you're asking whether your extracurricular activity (employment included) will be considered enough to counteract a lower-than-expected GPA, then the answer is likely No, as it's much easier to compare applicants using quantifiable values than in terms of time spent on non-academic pursuits.
Perhaps the more important consideration is how the outside activities and work influenced your development as a student. Some of these can already go on your CV, especially if your work has any relation to your studies at all. Sometimes it's enough just to prove you can accomplish self-defined goals or learn quickly. Any accomplishments you can note on your CV can be helpful. Noting that you've been part of a sports team won't directly account for a low GPA but it hopefully improved you as a person - if it has occupied such a large part of your time that you believe you would have had a higher GPA otherwise, then perhaps it's worth mentioning elsewhere in an application, such as the cover letter.
The extra-curriculars don't matter on their own, but if they're relevant to who you are, they may be selling points on a grad school application. Your accomplishments as a student will still be the primary focus.
See also this previous question: Do non-academic extra curricular activities matter for grad school applications?
What you're proposing is to write an application that says, "No, no, it's OK. I'm not an average student who got average grades: I'm an underperforming student who wasn't sufficiently committed to my course to get the grades I think I was capable of."
That isn't really an improvement over "I'm an average student who got average grades." Pretty much everybody who got average grades thinks they could have done better if only [insert some alternative history here].
The way to get somewhere that requires good grades is to get good grades. Find a master's course that will accept the grades you already have, fully commit to that course and get the grades you claim you're capable of. Then, apply for the next stage of your career (PhD, job, whatever) with the story of, "When I was an undergrad, I distracted myself with all kinds of things so I got average grades. But, look! Then I went to this decent university, fully committed to my course and got these awesome grades!"
Applications based on actual grades are strong; applications based on average grades and a promise to do better are, "Yeah, yeah, I bet s/he said that last time."