I agree strongly that math is poorly taught most places and that Concrete Mathematics does an admirable job of teaching math in a way that gets to the beauty of it. Oddly enough, it wasn't until I threw myself into more advanced mathematics that I started to see the fun in it. I absolutely hated learning calculus and linear algebra even though I could do it well. It just seemed so boring. I don't have a good solution to how to make things better, however, the mathematician Paul Lockhart has given this a lot of thought.
All that said, I think there are a number of misconceptions here.
First, mathematicians invent new languages all the time. That's the point of definitions, otherwise we'd be using sets to describe everything. The problem is, you first have to understand the concept well in order to apply suitable definitions. Think probability before Kolmogorov.
Second, Turing machines are a formalism to introduce you to the theory of computation because they are the simplest (or close to it) thing that can compute in the current sense of the word. Once you learn how TMs work, pretty much everyone just accepts them as a given and deals at a higher level.
Third people are trying to read Mochizuki's proof, but it is very hard. He basically invented his own way of doing things and so to understand his proof you first need to understand his methods. It's understandable that professional mathematicians with their own careers and areas of research find it hard to read the ~1000 pages of dense mathematics (proof + prior papers) to understand what is going on. Most people probably haven't read 1000 pages of math in their life, and it takes a while to come to terms with it no matter how smart you are.