Also I wonder if back then, there were whole cottage industries devoted to preparing people for these tests, like with the SAT or the MCAT today?
Yes, there were tutors and crammers. Even in my grant maintained Catholic primary school in the late 1960s we spent most of the final year cramming for the 11-plus exams so we could all get into grammar school. By the time we actually sat the exams we could do them in our sleep.
That cramming is how the school got its reputation and it's why parents wanted to send their children there. The downside is that I learned practically nothing during that whole year. I can still solve logic puzzles and parse English grammar like a champ though, in case the fate of the planet should ever hinge on that.