I actually learned this procedure in the mid-70's, probably the last generation to do so. But unlike long division or Euclid's Algorithm, I quickly forgot how to compute square roots since the process was quite cumbersome, one didn't have to take square roots that often and reasonably priced calculators appeared soon thereafter. Almost no one even 50 years ago used the manual method for square roots, using slide rules or log tables instead.
I rarely even use calculators today. I find square roots like I find out anything else, praying to the Google Gods.