I don't think your analysis is correct. Or at least, it's really incomplete and thereby misleading. I'm going to use hard numbers as someone who lives in one of the highest COL areas, but I want to emphasize I'm doing that for illustrative purposes and not to be condescending.
To begin with, the cost of basically anything you buy online from Amazon, Walmart, Apple, Best Buy, etc is the same no matter where you are in the country. Likewise for digital goods. That's a point in favor of the high COL areas.
Of course it's not that simple. You're right that there are plenty of things which cost more money in higher cost of living areas; namely entertainment, cinema, service-oriented experiences like restaurants, bespoke labor, groceries and housing.
In most of those cases the absolute cost raises significantly but the relative cost to your increased salary is still tiny; for example, I spend $6 - $8 for a half gallon of milk, but since I earn well over $300k/year that doesn't really matter. Similarly movie tickets are ~$18 but again, that doesn't scale enough to make much of a dent relative to a competitive engineering salary here.
On the other hand, some cost increases are significant even relative to competitive salaries. This mostly and primarily applies to housing, but it does also apply to restaurants and entertainment somewhat. But despite the fact that I spend over $4000/month for a luxury condo and another ~$2500/month on fun "stuff", I'm also saving over $100k/year on top of maxing out my 401k. That simply blows out any combination of lifestyle and savings I could enjoy in a meaningfully cheaper area.
Finally there is (unfortunately) an opportunity cost to working outside of high COL areas. The concentration of wealth and capital in high COL cities has a superlinear feedback effect on opportunity and lifestyle. There are numerous Michelin rated restaurants near me, a concierge and retinue of helpful staff in my building, world famous entertainment venues within a 20 minute train ride, numerous gyms, lots of childcare, excellent schools, etc. My commute to work is also only 20 minutes.
But those things don't interest everyone. More practically, it is also easier to quickly change jobs here, either out of necessity or for a quick 20 - 50% increase in compensation. Not only is the higher COL a justification for higher salary, but the employee power that comes with a bidding war puts a positive pressure on external compensation packages. The last time I went looking, I received about 10 offers. I don't even currently work at one of the most competitive companies according to levels.fyi.
I don't want to push this on other people because money isn't everything and it's perfectly valid to choose a lower COL area. But I do want to lay out the hard numbers from my experience so as to give a better picture for the situation.