Why does standard notation not preserve intervals (visually)
It does, but I think you are probably not accustomed to reading it, or how it was developed.
Let's first make an analogy with something familiar: reading English.
What is the meaning of "right" versus "right?" I can read the words, but only reading the single word isn't going to tell me the meaning. Of course we all know we need to read the context around the word. "His answer is right." "I write with my right hand." (Couldn't resist mixing it up more with a homophone.)
Following the analogy I can ask "why doesn't written English preserve the meaning of words?"
The answer is: "it does." But it doesn't preserve the meaning with single words. Complete meaning is only communicated in context.
Back to music notation...
Visually, you are mistaking this...
...to mean lines to spaces are fixed sizes of whole-steps or half-steps.
That is not how it works. The line and space represent steps on a diatonic series. The diatonic series is an asymmetrical series of whole and half steps represented by letters
A-G and the staff lines and spaces do not have fixed letter identities.
You must use clefs to know the letters of the staff lines and spaces and consequently the size of intervals between staff lines and spaces...
...where the middle line to the space above is
B to C a half-step or...
...where the middle line to the space above is
C to D a whole step.
I suppose the use of clefs is already understood by the OP. The question instead may have been "why use a system of clefs to know what the intervals and letter names of the staff?" It does seem to be a confusing system.
I think the reason for all the clefs is historical in two ways: 1) it evolved from notation for a single melodic vocal line (plainchant), and 2) it is harder to read ledger lines than a 4 or 5 line staff.
It's interesting to add that back during plainchant (Dark ages, Medieval era) there wasn't even a notion of absolute pitch for the tones of the staff. The tonic - or the selected range - was sung at whatever pitch was comfortable for the singer. A lot has changed with notation!
Depending on the range of the instrument a clef is selected that will put most of the notation on a 5 line staff, the clef gives a reference pitch like
F and that let's you know where the diatonic whole and half steps lay. The system isn't immediately intuitive and requires study. That's the reality. You can look for some alternate notation system - they do exist. But you will need to learn that too, and there may not be much available music depending on the system. Going back to the language analogy you can compare to the idea of Esperanto versus English. Alternate systems might be more logical (debatable?) but not practical.
...It's unnecessarily difficult to identify intervals like fifths or thirds.
One little addition about that.
The interval number will actually stay the same when counting lines and space regardless of clef, key signature, or accidentals. Ex. if you are on a line and consider that starting point 1 and then go up space 2, line 3, space 4, line 5. We have ascended to the 5th step and the interval number is a fifth,
5. Sames goes for spaces, but let's just stick to an example with lines. All of these are fifths...
...even with the various key signatures, placement of notes within the key, and the crazy accidentals of the third example they are all fifths.
Why? Because the lines and spaces represent the diatonic letters and we count those steps to first get the interval number:
E to G,
G to D,
B to F. Count them all on the letter series and you will see they are all 5 steps.
After the interval number we determine the interval quality. Those qualities are major, minor, augmented, and diminished. You can also have double diminished, and other complex intervals. At this point you do need to read the key signature and accidentals to know the specific quality.
But, rest assured that a fifth is always a fifth - of some quality - by the staff/letter counting. The three examples above are...
E to Bb a diminished fifth
G to D a perfect fifth
B♮ to Fb a double-diminished fifth
dd5 (sorry, that's really ugly, enharmonically it's a perfect fourth! I did it on purpose to illustrate the point, it's not very normal)
The same idea will apply with all the other intervals.
Key signatures with many sharps or flats and highly chromatic chords are difficult to read. The resulting enharmonic spellings like
Gbb which is enharmonically
F♮ are difficult, no doubt. But that is the extreme end of difficult notation.
If you really want to work on reading skills, work up gradually from simpler music. Mozart minuets or Czerny's Recreations are a good starting place. Simple key signature, mostly diatonic harmony. Schubert's various dances for piano would be a nice step up in complexity.