I wouldn't say lightning calculations but I absolutely do use algebra, as he does, to do rapid calculations and simplifications. For example, when he squares 103 he is clearly doing 1002 + 2x100x3 + 32 and the reason it's easy is because you don't even need to do any carrying. I don't make it a huge part of the class but I do show it when I'm teaching basic math classes for STEM majors, as well. A lot of our students will transfer to schools where calculators aren't allowed in math classes so they need to have a reasonable competency in arithmetic.
I agree with you completely on the topic of whether people can acquire it. Precise lightning calculation isn't as necessary anymore but being able to quickly whip out a reasonable estimate OR to know "wait a minute, that can't be right" when your calculator spits out something unreasonable are hugely important skills, and most people (barring specific math-related learning disabilities such as dyscalculia) can acquire it. Unfortunately a lot of people have such bad experiences with math in elementary school that it really puts them off the topic
On a side note, Cheaper by the dozen is an absolutely fabulous book, as is the sequel, Belles on their toes. I recommend both of them highly. They are somewhat dated as you might expect (given as the majority of the events took place before the first world war) but hold up much better than other books written about that time. My mother (who also loved the book as a child) mentioned that the first time she went to the Society of Women Engineers headquarters in NYC and saw a picture of Lillian Moller Gilbreth on the wall, her jaw dropped and "wait, she's REAL?!" Frankly, I find her even more interesting than her husband.