I'm with you on the not interrupting the flow of ideas. I am sure there may be composers who think about music from a different perspective and might find it easier to sketch out their ideas on paper. Maybe that's the way their ideas flow. I admire folks who can start writing notation (using a computer these days) and know what it will sound like before they play it. I'm not one of those. And I don't plan to start. It's not the way the creative part of my brain works - at least as far as music goes. I do like to write down my lyrics so I can look at them as I create and expand upon them.
The way I look at it, music is something you "listen" to. You can't really "read" music in the way you read a book. It is meant to be heard. So I think there is a great deal of value to composing on an instrument - and piano is certainly one of the more versatile instruments to compose on.
I have successfully written many complete songs, composing the melody, arranging the music and writing the lyrics. I compose on guitar (although I would love to improve my piano playing skills so I can also compose on piano). My definition of composing is that I sort of "hear" a melody in my head and I find the chords that go with that melody and then I write down the chord progression.
Disclaimer: I write country, folk, pop, blues and maybe some low key rock'n'roll. Classical music written for an orchestra may require a more academic approach.
Most of the time I have written some lyrics as I am composing the melody and I may write out the lyrics on a piece of paper and space them according to the meter and rhythm and phrasing of what I hear for the music and how that fits with the lyrics. Then I play chords to confirm that they match the melody in my head and once confirmed, write the chords above the lyrics where they go.
Once I have established a basic chord progression for the chorus and the verses, I may write the rest of the lyrics and decide if I need a bridge in which case I throw in a variation on the chord progression leading back into the final chorus. In most cases I try to make my verses and chorus different. I might use a pivot chord to switch the order of my chord progression - even if I use the same chords.
Before I attempted to compose music, I listened to music. I listened for many years starting at the age of 1 month or so. By the time I was ready to write my own music, I had a large database of musical phrases, melodies, rhythms and chord progressions in my brain. As soon as I learned to tap into that database in my head - and extract useful information that I was able to piece together into a song, I was a songwriter/composer.
It wasn't until after I started writing songs that I even learned much about theory. In the beginning I just knew that if I put a Cmaj, a Dmaj, and a Gmaj in my chord progression it sounded good - like some of the songs in my memory banks. It wasn't until later I learned that G C and D were the major chords in the key of G and that if I was in the key of G I could also use an Em and a Bm and an Am and they would all work.
As far as memorializing my melody - after I write out my chord progression and play through it a few times, I record a guitar vocal on a digital recorder in mp3 record mode - and then store the file on computer. After tweaking it for a few days, if I decide it's a keeper, I will use my Boss BR 800 multi track recorder to lay down a drum, guitar, vocal, harmony and lead guitar and if I get really ambitious a bass guitar. I can then finalize and master the recording after mixing it the way I like it, and convert to an Mp3 file and put it on my computer and e-mail it to my friends. I can even send it with a word document of the lyrics to the US Copyright office and receive a certificate of copyright.
The only music notation I know how to write is chords and that's pretty easy. If I want to notate a C chord I write "C" (imagine that). If its a C minor I write "Cm". That's about as complicated as it gets for me but it's all I need.
If I record a song in the studio, I let the session musicians listen to the basic guitar/vocal prior to the recording session and come up with their own parts to play over the vocals and to play during any solos. They never require anything more than a lead sheet with the chords written above the lyrics so when they listen to the guitar/vocal they can see what chord I'm playing at any given part of the song.
One thing I have started to do more of, is to analyse music that I like to try to understand what it is about the music that I like. Is it the rhythm, the chord progression, the range, the instrumentation, the structure? And then when I find a formula I can replicate, I use it to write an original composition of my own. That's as analytical as I get on the music part.
So I suppose that you might say that for the most part, I just feel out the music at my instrument. There is more of an academic process to writing the lyrics, but that's another subject for another day.
If you are more of a visual type thinker, there might be some value in learning to write out the music as you compose so you can see patterns and apply "theory" to exploit those patterns into something unique. But I am going to spend whatever time I am able to devote to music - feeling out and creating more original music. I'm too old and life is too short to go back to school.
There might be some related and useful information in the answers to this question Use of music theory in composition. And if you have an interest - here is a more detailed explanation of how I "write" music when I compose: What I do instead of transcribing my music into notation
Have fun composing, no matter what methods you employ.