When you began to learn to read, you would do it one letter at a time, and one word at a time.
"Tuh Huh Eh -- The ...
Cuh Ah Tuh -- Cat ... The cat ..."
... and so on.
As you improved, you'd speed up. You'd begin to recognise whole words at a time, then whole phrases. Now you can look at a page of writing and read it aloud at normal speaking pace.
If you learned a new writing system - say Cyrillic or Japanese Hiragana, you'd be back to the letter-by-letter slow reading style, but with applied practice, you'd improve and you'd be able to read Russian or Japanese text fluently.
Music is just the same. You are currently at the note-by- note "C.A.T. : cat" stage. If you practice, you will improve. You'll recognise (for example) a D, F#, A triad on the stave, and just play it. You'll recognise common rhythm patterns, and be able to play those too.
Like reading text, there's a sliding scale of complexity. A reader at a certain level might have no trouble reading Dick and Jane, literature aimed at children or tabloid newspaper articles, but would struggle to read an academic paper or James Joyce's Ulysses.
It's just the same in music. At the Dick and Jane level (for piano), would be a melody line with no accompaniment, in C major with no accidentals. Moving to a key with sharps and flats, a piece with accidentals, a simple left hand part, more complex rhythms and chords, and you make the job harder.
Different people have different preferences. I am not a fluent sight reader. I'm not even a particularly competent reader of sheet music. But I can play tunes by ear, and improvise chordal accompaniment.
My mother has no idea how to play a tune by ear nor construct a chord sequence to accompany a tune. She doesn't learn pieces from memory. But give her some sheet music, and she'll play it by sight.
I have a feeling that my mum represents the majority of "traditional" pianists - the people you find accompanying school choirs and playing church organs all over the world.