For me, reading music is like reading words: I cannot NOT do it. When you see printed words in a language you know you automatically read it - like it or not. Same with music - when I see it I "read" it in the sense I translate it into sound. Timbres, rhythm, volume, melody - all of it. Harmony can be a bit trickier no doubt. But most 18th, 19th c scores present no trouble at all anymore. It's a skill that takes time and training for most of us. Some music schools offer a course called "Score Reading" where you also learn to play a full score at the piano. It's hard.
Let me just give you one reason why having this ability is important: there are many opportunities for composers to write music for a competition of some sort. The composer submits a score. Then the judges have to read through them to decided what's good and what's not. They don't get to hear it performed, usually don't have recordings, and no one wants to hear a computer generated version. The judge must have a fine ear and be able to internalize what they see.
And yes, with enough experience, practice and talent, you write music on paper (or computer) without needing to play it on piano or something. You hear something in your head and just write it down with minimal difficulty.