Chess was mostly a game of leisure through the early 1900s. By this, I mean that most of the strongest chess players had a day job, or at least the appearance of one. They mostly learned by playing other strong players. So if we take Paul Morphy, undoubtedly the strongest player in the world in the 1860s, he learned by playing his father and uncle, then the strongest players he could find in New Orleans, then the United States, then in Europe. He would hang out at the Café de la Régence in Paris and play all day, every day, for months on end. As there was no lasting money in chess, Morphy would return to New Orleans, where he worked as a lawyer.

As chess was a public activity, and many considered it strange to study or play by oneself. This sentiment would continue well into the 20th century, when José Raúl Capablanca (world champion from 1921-1927) would famously claim that he didn't study, nor did he even have a chess set at home. Capablanca ostensibly worked as a diplomat for Cuba. So what changed? Well, you could argue that Alexander Alekhine (1892-1946) was the first to dedicate himself to chess as a profession, earning enough from tournaments and exhibitions to support himself. He was a voracious reader, and had strong memory for the games of other players.

But by far the biggest figure in the teaching of chess has to be former world champion Mikhail Botvinnik (1911-1995). After retiring from competitive chess, Botvinnik devoted himself to the nascent field of computer programming (he had a PhD in Electrical Engineering) as well as teaching the next generation of Soviet Champion. His famous school in Moscow produced most of the world champions of the late 20th century and early 21st century: Anatoly Karpov, Garry Kasparov, and Vladimir Kramnik.

Now you asked what people taught, but I had to give you significant backstory because his method was dramatically different than what you describe for your kid. Botvinnik believed, first and foremost, that learning occurred through playing a game, then annotating your own game, then reviewing your own annotations with other players/trainers, then learning from your mistakes. Furthermore, Botvinnik emphasized studying games from grandmasters, physical exercise (cardio), focusing on depth instead of breadth in opening repertoires, and simulating playing conditions (playing in loud and/or smoky rooms if that reflected the tournament venue). For his playing prowess and emphasis on training, Botvinnik is often called the Patriarch of Soviet Chess, and is cited by many players as hugely influential on their development.

Few active players remain from the Botvinnik School era (Peter Svidler is the notable example). Nowadays, top GMs spend a significant amount of time studying/analyzing games, starting with their own. But they've also supplemented this with training and exploration using technology. Top pros don't often talk openly about this, as it could help their opponents, but there is a push to use the best engines on strong machines (think supercomputers) to explore ideas in ways which could not be done previously.

If you want my opinion as a chess coach, chess is about pattern recognition, and the better you remember patterns, the better you will do. The reason Magnus Carlsen is World Champion is because he has a phenomenal memory for games. He can recall over 10,000 games that he has played and studied, and recognize them from nearly any position reached in those games. It's easy for me to state all of this, but let me give you a playlist to demonstrate it:

My presentation on Paul Morphy. Search YouTube for "Life and Chess of" for Capablanca, Alekhine, Botvinnik, Kasparov, Karpov, et al. I give my book sources at the end of every presentation.

2011 Feature on Magnus Carlsen where he demonstrates his memory

A very recent video where Carlsen demonstrates his intuition for endgames