This. Very much this.
Everything in this world is pretty much built upon axiomatic systems. The language we speak, the laws that govern us, economics, etc. It's going to be quite hard to spend one day without interacting with at least a dozen or more axiomatic systems. Without even a cursory understanding of how rules and these systems behave, how can we hope to interact effectively with them?
So that brings up the question, what's the easiest and clearest axiomatic system that we encounter on a day to day basis? Mathematics. The study of math isn't so much so that students learn the principals so they can apply them in during the rest of their lives (although there are some that they do need to know), it's so they can better understand how rule based systems operate. Therefore there's no real need for a literary student to be able to prove the Quadratic Formula. What IS important is that they know a proof exists, and they understand why it can be proven. That's the main key.
So yes, trim calculus for most students. Trim formal proofs from tests (but do demonstrate them in class). Focus more on the why, not the how. Then once that's nailed down, go on to the topics that are of real value (but are often under-taught IMHO): basic statistics and probability, formal and prepositional logic, and mental estimation... Each one is used nearly every day by people whether they realize it or not. It doesn't matter that they know that If P then Q; P; therefore Q is called modus ponens, but they must know that it does work. They don't need to be able to determine a probability of an even happening, but they do need to understand and know the basic concepts.
I will disagree slightly with your statement: With calculators and computers, nobody needs to know math itself.. The correct (IMHO) statement would be that most people don't need to be able to prove what the calculator is giving them is correct, but they do need at least a cursory understanding of what's going on, otherwise how will they know if it's correct? If you punch 5 + 10 * 100 into a calculator, you'll get back 5000 on a stack based calculator (computer) but 1500 on a normal calculator... If you don't understand what's going on, how do you know which is right (or that there are two possible answers)...?
Again, just my $0.02