I agree with the sentiment of the OP.

For an answer, there have been various influences:

(1) Whatever math was before 1900 or so, by the time of the Russell paradox and its fix with axiomatic set theory, the style of the fix was to be close to Russell-Whitehead (if I have that right) idea that proofs could be checked essentially mechanically by just symbol substitution and manipulation. E.g., in those days there was a book on the natural numbers, that is, 1, 2, 3, ... that apologized for numbering the pages before the natural numbers had been carefully defined!

There was even a name given to this style of writing, telegraph style.

(2) Of course, the books written on axiomatic set theory easily fell into the telegraph style. Even there the writers were getting into trouble: They gave names for the various axioms; they didn't explain why the names they gave were appropriate, and I never could discover why. But I was eager to get out of the sub-sub-basement of axiomatic set theory ASAP so just did a f'get about it.

(3) When books were written on abstract algebra, e.g., basic set theory, construction of the main number systems -- the naturals, integers, rationals, reals, complex -- and then went on to the main algebraic systems defined with axioms -- groups, rings, fields, vector spaces -- it was easy to stay with the telegraph style. E.g., it was tough to find a book on abstract algebra that also discussed group representations and its applications to quantum mechanics and molecular spectroscopy.

(4) Long calculus was often done with a lot of intuition and nothing like some carefully done definitions, theorems, and proofs as in, say, W. Rudin, Principles of Mathematical Analysis. And physics and engineering kept drawing diagrams with "little interval dx" etc. So, when abstract algebra was proving theorems, the calculus authors also wanted to be careful at least about delta-epsilon arguments. Then including a lot of physics, mechanical engineering, touching on the heat equation or fluid flow, was considered off-topic. Bummer.

(5) The series of astoundingly carefully written books, close to telegraph style, by the team Bourbaki was influential.

(6) During the Cold War and the Space Race, US math was awash in grant money and essentially turned its back on the physical science motivations and applications. Some of the funding people started to get angry about that, and we got the Tom Lehrer song and joke about abstract math being about "the analytic algebraic topology of locally Euclidean metrization of infinitely differentiable Riemannian manifolds" or some such.

But, sure, especially in analysis, for a good proof, there's often a good picture and if can see the picture then can construct the proof easily. E.g., for positive integer n, the set of real numbers R, and convex f: R^n --> R, that f is continuous has a really cute picture. Same for Jensen's inequality. In linear algebra, the polar decomposition says that each square matrix is just (A) a rigid motion, rotation and/or reflections followed by (B) moving a sphere into an ellipsoid by stretching and/or shrinking on mutually orthogonal axes. One or more of the axes goes to zero if and only if the matrix is singular. Etc. Nice picture.

Currently, then, there is an opportunity for math authors to include motivations for their subject, definitions, and theorems, intuitive descriptions and helpful pictures, applications, issues, open questions, etc. Uh, when reading a proof, for each of the assumptions, check off where it was used in the proof! The definitions, theorems, and proofs can still be fully precise and solid.

Some students of, say, analysis have long tried to find and draw pictures that would clarify what was going on in some definitions, theorems, and proofs. I would advise new students to do that also.