Having had to address the issue of reading in recent years:
First - there is READING - before there is sight reading ( meaning prima vista which is a specialized skill needed by accompanists and such) - and what it entails.
In my personal journey, I got at certain aspects of it from an odd angle, which is in fact useful but incomplete. I had no lessons for almost 50 years, but in grade 2 one teacher decided to give us movable Do solfege. From then on I related music to the major scale and relative natural minor scale, as though moving along a ladder of unevenly spaced rungs. -- Next I was given a relative's book which had sonatinas, esp. Clementi, and one of Czerni. This music is very diatonic, and Clementi uses a predictable structure which is why a lot of actual musicians find his music boring. Clementi does the I IV V thing along a predictable Alberti bass. He climbs up to the Dominant key through some sequenced scales, repeats a similar theme, then climbs down again and finishes. Through this diet of Clementi in my self-study as a child, I picked up basic "classical" structure.
A child will pick up grammar and syntax before ever going to school, and use these instinctively. It was similar to that. Also - think of Knock Knock jokes - there is a pattern. If someone says "Knock knock" you know you have to say "Who's there" and you can predict he will say some new word like "Banana" and then you have to say that word plus "who" and so it goes. In this way I was able to anticipate where the music would probably go, and then look for clues whether it was doing that.
Take for example the rounded binary pattern - C major, G major, C major. You'll see lots of sharps for the F# in the middle, and you can see that in a glance. Take a scale - it's a slanted straight line without accidentals if it is in the same key and major.
This was the kind of "reading" I did. You can predict where a knock knock joke goes, but if a person is conversing and happens to say "knock knock" you won't be able to predict it. If the music stops following those patterns, stops being diatonic, then this kind of "reading stops working. But there is an element of reading in this. This goes with what Michael wrote:
If you have a good understanding of the musical style of the piece, you will know to look for the end of a phrase, or the end of a rhythmical or melodic sequence.
I was at home with the form of sonatinas, especially our predictable Clementi, and could do a lot of predicting.
There are other aspects of reading. One that is specific to piano is the ability to associate a symbol in the score with a location on the keyboard. This can become as automatic as seeing a red light and your foot goes to the brakes while you anticipating slowing down - it is one single thing. I spent several years acquiring this skill, because I did not have it at all. My relative Do solfege left me with a lot of holes. Music is also a physical thing, and in this sense it is more like dance or gymnastics than it is like language.
Another aspect again is understanding theory, but not as something memorized from books, or symbols moved around on a page like a mix of algebra and geometry - it has to mesh with sound and music. Simply grasping a key signature and time signatures and being able to relate them to your instrument are part of learning to read. People talk too much about "sight reading", and being able to go at tempo etc. as though they were training to become accompanists, before just reading is mastered. Here I think there are underlying skills to be chased.