In regard to whether real analysis makes "water-tight" proofs possible, I would say that even that statement is highly debatable. In any scientific endeavor, what is considered as "water-tight" today often becomes not so after some time. In fact, Henry Poincare made that observation in one of his speeches. How do we know for sure that we have not made some kind of implicit assumptions based on what we currently consider as "self-evident"?

Generally speaking, what is considered as a valid proof is driven more by the fame of the one who produces the alleged proof than by mere merit. Take, for example, Any Wile's "proof" of Fermat's Last Theorem. Have you seen it and understood it yourself? If not, why do you accept that it has been proved? It seems to me that, just like other things in the human world, what is "true" is a matter of social consensus. In this case, it is more like the lack of dissent opinion from a very small elite group of experts. In the history of math, there are many math geniuses who did not water-tight proofs or produced proofs with errors. For example, Riemann's proof of his mapping theorem was known to be defective. It was not cleaned up until many years later. Yet, the idea was correct. In addition, it is commonly known the Euclid's work is full of holes. Yet, that fact has not prevented him from making an important contribution to mathematics. Thus, it seems that the importance of formality in general and of real analysis in particular has been oversold.

It seems that our basic disagreement has something to do with your unfamiliarity with the history and philosophy of mathematics. Many people think of math as the last bulwark of certainty. It is not. Math has lost its certainty since the emergence of non-Euclidean geometry. I tend to agree with you that if you have an axiomatic system and you deduce your conclusion through careful logical reasoning in every step, then you can achieve a high degree of certainty. But what about the assumptions of the system? By definition, the axioms are assumed to be true. They are not subject to tests.