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I agree that the most important thing is convincing the 1 person out of 1,000 to be a math major (or physics or whatever) I'm afraid the university isn't ok with handing everyone else As without mastering a certain minimum skill set.
Besides, the math department isn't a free agent in this. Other departments expect students who have passed math 1A to have certain skills. If the math department was free to teach math 1A as an advertisement for mathematics (the way other departments get to teach their base courses) it would be a much different class. Hell, it wouldn't even be about calculus.
So long as the physicists, economists, chemists etc.. etc.. say "we need students with skills X, Y and Z" there has to be a class whose content (and hence a decent grade in) is X, Y and Z. The name of that class is 1A and it helps no one to just teach a different class under that name without reforming the system.
The math department ALREADY OFFERS OTHER CLASSES DESIGNED TO DO EXPOSE STUDENTS TO MATH IN EXACTLY THE WAY YOU MENTION. These are courses that deliberately avoid the equations and rote memorization that generated so much fear in high school and are designed to be low stress while showing off the wonder and creativity of mathematics.
Unfortunately, few people ever take these courses because students don't want any more math than they have to take and the guidance counselors and university won't push or require them.
As you said about the final (which I wholeheartedly agree with) if you give students the choice how is it the department's fault you take it. Students have a choice to approach math in a more exploratory, more encouraging fashion but they choose to do the minimum needed to take whatever other class they need. It's ironic because that's also what makes calculus so hard for them...they want to apply a rule and confidently move on...not think hard about a math problem in ways that might be dead ends.
That is what's so sad about math education. Understanding comes from trying to fit the concepts together and even (especially) professional mathematicians fail 90% of the time. Students, especially those who haven't done well in math find that very distressing. Hell, pick any subject you aren't good at (which has clear success/failure conditions) and really try to do it well. It makes you feel dumb and you give up. Add to that years of pre-college math (often taught by teachers who themselves seek the reassurance of rote procedure) which brain washes the curious kids into believing that doing math is just applying procedures and it's an uphill battle.
Ultimately, I think the problem is that we try and force kids who aren't really interested to know math they don't need. Just like other subjects let the students who want to learn come to you. With the advent of modern computer algebra system no one benefits making kids memorize rote procedures beyond basic arithmetic since without understanding they will never apply it in the real world.