I never bothered memorizing stuff which is "fragile", e.g. browser quirks. Luckily, there's no shortage of programming knowledge that is the very opposite of fragile. Under a few common assumptions, sorting can't be done faster than O(n log n). All NFAs can be converted to DFAs. 2-SAT is polynomial but 3-SAT is NP-complete. And so on.
People who disdain math as "brain teasers" are deprived of the rush of knowing that some piece of your knowledge will stay true forever.
You can't get useful work done on real problems without learning a lot of that "fragile" stuff. How are things scheduled on your OS? Where do you find your logs? How do you run unit tests? What are your local coding standards? What is the library call for doing X? How do you find your documentation? What was that section of code you found foo in the other day?
All of this stuff is fragile. None of it is going to last. But it is context for your current life, and you're going to be more productive if you learn it. And when this stuff changes on you, you're going to feel the productivity drop. But there is no sense in hamstringing yourself so that you won't feel so bad later. You learn it now, and you learn the replacement later.
Back in the 1800s there was a belief that you could only learn so much, so you had to keep your brain clear for the important stuff. (Go read Sherlock Holmes for an example of this prejudice.) But these days we know better. People can practice learning. And those who do, constantly are learning. About everything. Including trivia. And it pays off. It really, really does.
People who disdain math as "brain teasers" are deprived of the rush of knowing that some piece of your knowledge will stay true forever.