Hi, there. Professional mathematician here. Sorry to hear about your experience. In truth, I believe you. That said, I believe there are some historical and pedagogical reasons for your experience.

1. Often, math is used to teach skills outside of their mathematical usefulness and I wish we'd be more honest about it. For example, take the controversial topic of long division. As a professional, I never use. Ever. In fact, we use a different algorithm computationally on a computer. That said, why did we teach it? It's one of the first algorithms that students learn. As such, it teaches organization and a methodical following of steps. Now, is this the only way to teach this? Of course not. In years past, we taught square roots. Candidly, a similar sort of precision can be taught with computer programming. However, long division can also fulfill this role even though, again, it's practically not that useful.

2. As far as proofs, you're right. It's just a complex argument that something is true or not. In fact, it's an incredibly flawed process as well since spoken and written language tends to lack the precision to be absolutely sure. Now, there are very formal ways to prove things using techniques from, for example, the mathematical logic community, which can be computationally realized in proof based systems like Coq or Isabelle. However, this is hard, so no one really does it. As such, why do we stick with a possibly flawed proof system that's a pain for most people?

Well, I'm sure there are lots of reasons, but the big one for me is that there many circumstances where intuition breaks down. In my studies, the first big breakdown in intuition occurred during calculus and the first experience with the infinite. Another big breakdown occurred in the transition between real analysis (calculus) and functional analysis. For example, the unit sphere is compact in finite dimensions, but not in infinite. It just works differently.

Now, does that mean that intuition isn't good or used? Of course not. However, the proofs help bring forth the brittleness, or robustness, of a situation by systematically breaking down where things work or do not work. For me, it's a way to add structure to a problem that I'm working with in order to ensure I know what's going on.

OK, so I'm a mathematician and my interests may be different. However, think about it from an engineering discipline. We can model things like fluid flow or electromagnetics and achieve really good, useful results. Most of the time. All of these equations have assumptions behind them and when these assumptions are violated, everything breaks. For example, do the governing equations and algorithms work when the domain has a reentrant corner?

As another example, we use optimization solvers in many domains from engineering design to machine learning. The equations that these solvers use for constrained optimization depend on something called constraint qualifications. When these qualifications don't hold, everything breaks and we don't find solutions. For me, the constraint qualifications such as when the tangent cone coincides with the linearized cone (Abadie CQ) aren't particularly intuitive. It's an artificial construction that results from the proof of equivalence between two formulations. However, it's also essential for the algorithms to work.

Anyway, none of this is meant to refute your experience. I completely believe you. Really, it's a way to provide some clarity as to why some of these things are taught in this way and why they may be valuable for other reasons. I do believe that math can be taught better. I also believe that there's not a universal way to teach math and that different approaches resonate with different people.