Not really. It's a suggested bit of advice in the manual for building worlds to use as a state to work from. It's the basic world, upon which DMs are expected to add if they wish. The actual rules of the game directly contradict it by being very free with magic, and by providing rules for the crafting of magic items.
The DMG unequivocally states that the DEFAULT assumption is that spellcasters are rare. Then it gives advice on different ways to handle magic. The default of D&D is to have rare spellcasters, though. So while yes, the DM can build off of that default and change spellcasters to be more common, if he doesn't, they aren't. No change = default.
Spell books aren't sturdy, first of all. Secondly, a person cannot just find an ancient master's spellbook (miraculously in a language they understand) and go from not knowing what an integer is to understanding calculus. Even Da Vinci, Newton, Galileo, all learned the fundamentals from teachers and from books made for the purpose of instruction. Third, if "person who can learn to cast literally any arcane spells at all" is as rare as you insist, then the few people who can are going to hoard their tomes against rivals, and people who don't want Mage-Emperors are going to burn the libraries of dead mages to prevent some young bastard from finding it and becoming the next best thing to a god.
Spellbooks are not written in languages. They are written in some sort of magical runes or something. We know this because wizards without a roll or spell can copy a spell out of literally every spellbook they find, no matter where in the world, or even found on another world in another universe. That would not be possible if they had to be able to read a language that they don't know.
It's exceptionally cheesy. It's cheesier than He-Man.
Prove that it's cheesy. Casters needing to be born with a rare talent for magic is a common trope in fantasy and in D&D.
And please, if you are going to insist that you're just pointing out the defaults, at least provide quotations with page numbers, or screencaps, or something else people can use to fact check you without having to dig through a book you insist says exactly what you claim.
DMG page 9 under Core Assumptions about the game world.
"The World Is Magical.
Practitioners of magic are relatively few in number, but they leave evidence of their craft everywhere."
DMG page 23 under Magic in Your World.
"What normal folk know of magic depends on where they live and whether they know characters who practice magic.
Citizens of an isolated hamlet might not have seen true magic used for generations and speak in whispers of the strange powers of the old hermit living in the nearby woods."
If magic were as common as you make it out to be, that bolded part would not be true.
DMG page 24 under the same heading.
"Consider these questions when fitting magic into your world: Is some magic common? Is some socially unacceptable? Which magic is rare?"
The reason you have to ask yourself if some magic is common, is because by default it isn't.