Why do we still do symbolic math? I just read that most practical problems (algebraic equations, differential equations) do not have a symbolic solution, but only a numerical one.

Numerical computations, to my understanding, never deal with irrational numbers, but only rational numbers.

Why do mathematicians construct the real numbers and go through the pain of using their complicated properties to develop real analysis, etc. to be able to solve just a few cases symbolically?

Why do non-mathematicians deal with real numbers and symbolic calculation if the majority of cases must use rationals and numerical computation only?
Try to solve $x^{1/3} = 0$ with a "symbolic calculation". Now try to solve $x^{1/3} = 0$ with Newton's Method. The point is numerical methods can fail and anyone using numerical methods should have an understanding of mathematics so that they can detect when a numerical method is not working and perhaps even fix the problem. Also as this case shows sometimes the symbolic calculation is easier.

All these computational methods need to be designed and implemented. It takes an understanding of the underlying mathematics to do this.
There are too many sub-topics in math but it is split with two main topic theorical and applied mathematics. You can't prove 2n is an even number with numbers it goes to infinity and a simple question like ' a car try to reach a target. There is a road between target and car about 100 km. car is reaching half of way every time. 50,25,12.5,... when it is end ? Can car reach the target ? You cant answer questions like this without symbolic math.As a mathematician I can say numbers is just seen side of iceberg