1st edition was a toolkit. It was designed to be heavily modded to suit individual tables. Hence the word "advanced" in the title.
Incorrect.
Gary himself said that the AD&D rules were supposed to be the definitive official rules for D&D rather than the loosey-goosey everyone-does-it-their-own-way rules Basic/OD&D ran under. Gary repeatedly talks about how you should be using every rule in the PHB, DMG, and MM (except for the ones explictly marked optional, like bards or psionics) and that anyone not using them isn't really playing D&D. That was evident in his DMG and in subsequent writing in Dragon.
Now, nobody ACTUALLY PLAYED like that. Mostly due to the fact that AD&D 1e is an absolute nightmare to play RAW. Even Gary rarely played according to the rules he himself wrote. Hence why most "AD&D" games were really a mixture or Basic, 1e, and later 2e based primarily on what the DM liked or remembered. The only people who tended to play BECMI or AD&D 2e exclusively are those (like me) who found the game in the 90s and those were the books that were for sale in an era before the internet made finding the old books easy.
And unlike 1e, AD&D 2e WAS a toolkit. There is a lot more explicit opt-in rules (such as proficiencies and individualized initiative). Even the class selection was a toolkit, hence why specialty priests are literally a set of (bad) DM guidelines and the only specialist wizard listed in the illusionist. It was often assumed you would use everything, but unlike Gary's "use everything or your not really playing AD&D" decree, Zeb Cook actually tried to make 2e modular. How much he succeeded is highly debated, considering how much of a redheaded stepchild 2e is still considered 30 years later.
Honestly, if I ever had the inclination to play AD&D 2e, I would probably try to build a Definitive edition using a mixture of the 2e PHB and elements from the PO line and other supplements to make a real "2.5" that fixes some of the issues with the 2e PHB. However, I'm pretty sure I'd have to play such a project straight as the minute I would want to start fixing my own issues, I'd probably just invent 3rd edition again.