I actually live in high COL and low COL areas at the same time, thanks to working far from family and friends. My high area is actually, as others point out, cheaper for everything except housing, thanks to scale and location and immigrants.
So it balances out, right? No, actually. I enjoy living in less house than most people, but that's hard to do in my "low" COL area, as there are far fewer small/rental units available, and most of those are bad.
So, as long as I avoid fashionable neighbourhoods, the "mustachian dividend" for my lifestyle choices is greater in my high-COL area!
Of course, I could optimize further by putting my high-COL place up on airbnb when I'm not there. Or sharing my too-large low-COL place.