If UberBlack is 2x the price you pay, what you pay is close to 2x what UberX costs (depending on the market).
It's fairly normal to take a taxi to the airport (particularly at your destination), but paying 2x the cost of a taxi for a private car service is definitely "luxury."
4x is not the ratio of the prices that I'm seeing from the fare estimator. Regardless, if you want to believe that I'm not normal, that's your prerogative, though I'm not sure what benefit you derive from insisting on this. Additionally I am highly suspect of your implicit claim that you're more "normal" than me if you regularly take Uber to and from the airport. I rarely even fly. If you fly more than twice a year and take Uber each time, you are spending more than me on rides to and from the airport.
Your choice of words is curious to me as well. For instance, you distinguish the limo service I use as being "private". What does that mean in this context, and how is Uber not private? You've misunderstood me if you think I have a personal chauffeur who exclusively works for me. That's not how towncar services work. It's basically just a guy and a few friends who have a tiny company where they drive people to and from the airport.
By the way, I want to point out that Uber's surge pricing could easily make UberX cost more than what I pay. And surge pricing is fairly likely to be in effect any time I'm trying to get to the airport.
It's fairly normal to take a taxi to the airport (particularly at your destination), but paying 2x the cost of a taxi for a private car service is definitely "luxury."