I think you are being unfairly critical in your first paragraph. You seem to have taken for granted the relative ease of access to certain ideas in mathematics that took many decades to develop. You say that intuition is often abandoned in the presentation of ideas (while unfairly generalizing this behavior to all mathematicians), but how hard have you tried to make the abstract proofs appeal to your intuition?
You seem to suggest that pioneering mathematicians constructed many geometric arguments to aid in their intuition, before presenting an abstract proof. Yet it is generally agreed upon that a diagram or picture does not constitute a proof. A rigorous proof ensures that our intuition is actually correct and reliable.
For instance, I'm not sure if you are familiar with the Three Hard Theorems from Spivak (IVT, A continuous function on a closed interval is bounded and also has a maximum/minimum value), but in each of these theorems, the geometric argument is quite clear, perhaps some would say undeniable for the IVT. My initial impression of the proofs (which are given in the chapter on least upper bounds) was dismissive. The mechanics of the proofs did not appeal to intuition and I thought they weren't worth the trouble.
However, I made an effort to gain a better understanding of the proofs (admittedly, about 5 days before the exam). I read the end of the chapter on the Three Hard Theorems carefully. There, Spivak presented the motivation behind a rigorous proof of the Intermediate Value Theorem. At the same time, he explained the necessity of the existence number that would bound all the elements of a set to complete the proof. Soon it became clear to me why the LUB property is an axiom. Afterward, the proofs of the Three Hard Theorems were much more clear. In fact, I think the proofs improved my intuition (except for maybe the maximum value proof in which Spivak used a trick to reach a contradiction using the 2nd hard theorem) rather than hindering it. They presented arguments that were not as far away from the purely geometric arguments as I had imagined.
Now perhaps the biggest flaw with my example is that Spivak does everything for the student. Although I'm not taking real analysis until next year, I'm not blind to the fact that certain real analysis books, if not all, will not have nice diagrams and geometric arguments. However, I have faith that it is still possible to obtain intuition and motivation in the geometric sense or otherwise. I understand that this will take more work on my behalf to fill in what has not been presented. My rationale is that from doing the problems in Spivak, whether assigned or not, I am involved in the process of taking an intuitive idea and making it rigorous.
I think my second example will address some of the other questions you raise. I am currently working through Chapter 13 of Spivak, the chapter introducing integration. I believe what you're referring to as "useless for application" is known as Darboux Integration.
As a high school senior last year, I took Calc BC. We proved almost nothing in the class. My intuitive grasp of limits and derivatives was gained in precalc, and somewhat reinforced in Calc BC. I really cared about how integration would be presented, since my senior friends who took the course the previous year all claimed that differentiation was much more intuitive than integration. The presentation of the theory of integration lasted maybe 10 minutes, probably less. Essentially, the teacher drew some rectangles on the board, threw us a definition, and never referred to the definition again.
Soon the FTC was introduced and they were a bit easier to grasp through concrete, numerical examples, although no proofs were given. I quickly learned how to integrate and the rules and tricks came easily to me. Eventually, I held onto the notion of the sum of differentials to aid my intuition, even though this concept did not affect my ability to integrate at all. I retained the idea to comfort my intuition, but I couldn't really use it to do anything else.
Although I haven't proved many of the chapter theorems using the darboux integrability criterion, I can already see why the definition is useful. First of all, it made more sense than the other definitions I have encountered. I mentioned that my Calc BC class threw a definition that was not really used. I soon looked through Stewart's and was somewhat satisfied with his presentation involving sample points and taking a limit. By then, I figured the concept of integration was something like "we add more and more rectangles while making the dimensions of the rectangles smaller to approximate the area".
Spivak's presentation reinforced my intuition. The base of the rectangle was determined by partitioned subintervals, the height determined by sups and infs of f on the interval. I learned what it meant to "add more rectangles" by understanding why L(f,P) <= L(f,Q) where P is a partition with a lesser number of points than Q (similar idea for upper sums). Then the use of sup{L(f,P)} made sense (and also inf{U(f,P)}. It was easier to see how adding more and more rectangles that become smaller and smaller captured the true area from above and below. The application of the definition to many simple functions convinced me of the definition's usefulness.
So is this definition more useful to me than the notion of "summing the differentials"? Until I learn the rigorous formulation of the latter process, yes. I repeat that viewing integration as summing differentials was of no use to me when I think about it. Now I have a definition that makes intuitive sense to me, can be used in key proofs, can be used to prove certain problems by itself, and probably used to numerically approximate the area under the curve.
I don't expect you to agree with the last paragraph, and that's perfectly fine. But to me, there is a very strong rational basis for sticking with the darboux definition, partly because of its practical usefulness. I find that my intuition is improved with the definition and that the definition itself is applicable.
On a final note, I suggest you try taking a look at Understanding Analysis by Stephen Abbott. Although it might not be as comprehensive as other real analysis books (indeed, it is an "intro to analysis" book), I would place it more towards "analysis" than "theoretical calculus with rigorous proof".
, but still it raises a question about syllabuses and teaching. 
