While I agree that education does needs some considerable improvements your claim that there's a need for a universal-experience class doesn't really hold water.
In school we learn wonderful things like how to find integrals, solve equations, and how to calculate valence electrons of elements based on their atomic numbers. Because, obviously, they will be very important in our futures -- especially if we become artists, musicians, writers, actors, and business people.
Ironically despite being intended as sarcasm this statement (sarcasm aside) is fairly accurate. However you're referring to the usefulness of being able to solve particular kinds of mathematical and scientific problems rather than the actual subjects in question. When you take all of the subjects into account rather than those cherry picked items you find that they are indeed useful for artists, musicians, writers, actors, and business people.
Just to start off with learning Physics is very different to learning (for example) the equation for calculating gravitational attraction. Physics is about being able to find the equation for gravitational attraction though observation and analysis without needing to be given it. That sort of analytical problem solving would actually be incredibly useful for someone going into business. Similarly the development of curiosity and imagination that should go hand in hand with learning physics is vital for the arts. The principles of resonance are actually useful to a musician for understanding why their instruments work the way they do. I'd go on but I'm sure you get the point.
Moreover the things you are suggesting should be taught in this universal experience class are things that would be already included in a proper education in the Arts, Sciences, Humanities and Mathematics. Leading me to suspect that the problem lies in that your education (like my own) has been found to be lacking these core skills. The problem leading to things being so badly taught is that they have very little bearing on the actual exams, you don't need to be able to derive the general derivative of x^n to be able to take the derivative of x^2. So such things get left out in the race to make sure students pass their exams.
Creating a universal experience class that wouldn't end up being reduced to memorization (somewhat like what has happened to philosophy these days) would require you to solve the problem outlined above; and if you can do that then it would render the universal experience class obsolete. Seeing as you could just apply that solution to all of the other classes and cover all the ground covered in the universal experience class and more to boot.
On a side note I'm curious as to what you mean by "everyday thinking"?
In school we learn wonderful things like how to find integrals, solve equations, and how to calculate valence electrons of elements based on their atomic numbers. Because, obviously, they will be very important in our futures -- especially if we become artists, musicians, writers, actors, and business people.
We learn so much in school. Yet, when most people look at paintings they don’t truly understand them. When most people listen to music, they don’t really know what they’re hearing. Most people would fail simple music theory tests, even though many have listened to music most days of the week since they were babies!
Similarly, if you have working eyes, you should ask “Why do shadows look like they do? What color is snow, really? Can I predict the colors of different colored materials at different times of the day? If not, why? I have been seeing them for years, haven’t I?”
I think the problem here is that people can’t understand what is really important. Calculus, mechanical physics, chemistry, microiology, etc. are interesting to learn, perhaps. But, they are relatively advanced topics. People don’t use them in daily life unless they are professionals. Why not learn things that we think about every day instead of those that will frankly be useless to most?
Why don’t we learn how to understand our senses?
Learning about sight, sounds, thoughts, etc. should fit in somewhere in the first year of high school. Everyone needs to learn the physics of art and color (e.g. this and this), music theory, rationality, and logic.
For example, why should people start learning (or pretending to learn) philosophy, the art of thinking, in college? Should we be able to make life-changing decisions without even knowing how to spot errors in our thinking?
As a science researcher, I know first hand how hard it is to find a good balance between being well versed in worldly topics and being focused on a field in order to excel in it. But, both of these areas of study should not be called the true basics, in my opinion.
As president of my school's philosophy club, I took a different approach to teaching the basics of philosophy and thinking than traditional classes do. Instead of asking students to discuss the lives and ideas of famous Greek philosophers, I asked them to analyze their own lives and make their own philosophies. As expected, they were terrible at it at first. But, by the end of the year people began to actually think about the world around them.
So, my point is that we should -- in life and in school -- emphasize actual everyday thinking more.
The biggest challenge is that it takes so long!