Yet Russian people learn the language just as OP did. Those tables serve at best as references to be consulted rarely, not as an organic learning material, that would be insanity.
They do, they start as toddlers, they practice the language 24/7, and they have a decade to master it. Adult people simply don't have this luxury. Kids are not really a model for learning as an adult.
Learning is an effort that needs to be put. Preferably, in a time frame narrow enough you don't forget why you even started learning. And when optimizing the language learning process, grammar tables are useful.
I have learned Russian at around 10 years and Engish at 22 and in both cases grammar tables and rules were waste of time. After accumulating some initial vocabulary, the best route was to read books and watch tv.
A lot of time in school is spent learning grammar. And one (small) part of it is conjugation - you actually learn the rules. For situations where feel guides you wrong. Or for arguing - if there is disagreement you solve it y rules.
Learning is an effort that needs to be put. Preferably, in a time frame narrow enough you don't forget why you even started learning. And when optimizing the language learning process, grammar tables are useful.