You're not teaching them to doubt a source. You're teaching them to use their brains. If you're teaching by rote, chances are (though not always), you're not teaching much of anything. Multiplication tables are garbage, for instance; math is not about being able to calculate random garbage in your head quickly, but even if it were, people will naturally memorize things they see often.
"Critical thinking is a way of deciding whether a claim is true, partially true, or false. Critical thinking is a process that leads to skills that can be learned, mastered and used. Critical thinking is a tool by which one can come about reasoned conclusions based on a reasoned process. This process incorporates passion and creativity, but guides it with discipline, practicality and common sense."
The problem with "teaching critical thinking", then, is that you teach them to decide validity for themselves, but you do NOT teach them to always come to the correct conclusion when the process is complete. If it were about coming to "correct conclusions", rather than "conclusions which appear on the surface to be correct based on the available information", everyone who applied the critical thinking process would always arrive at exactly the same conclusions.
Therefore, you want to maximize "the available information", and turning on the write protect bit on someone's mind before they understand, for example, that "being able to calculate random garbage in your head quickly" is *very* important to knowing whether you have enough money in your pocket to buy the things you have in your shopping basket when you get to the checkout, or whether you're going to look like an ass while you make everyone behind you in line wait while you decide which things to put back to fit your available cash.
is effectively cultural genocide.
Education is not about skill sets, or keeping certain cultures that rely on obedience alive.
I think you really need to look up the term "Cultural Relativism"; not everyone wants to live in an apartment with cable TV, within walking distance of a Walmart, a McDonald's, and a Pizza Hut. Other cultures value other things, and they keep their cultural integrity, often by operating on a different axiomatic basis that has to necessarily exclude some information, but which does not preclude critical thinking within their (different, not necessarily limited) scope.
There are cultures which achieve what they believe to be very fulfilling lives in that way, and tend to have vastly lower suicide rates than our supposedly "superior" culture has; your statement is tantamount to another term you should look up: "Cultural Imperialism".