I think part of the issue here is that you’re comparing ASL - which has several non-concatenative (I.e. simultaneous) elements - to spoken languages, which have more limitations as far as that’s concerned and are typically concatenative (I.e. linear). Phonologically, a sign has 4 elements: handshape (the configuration your fingers and hand make), movement, palm orientation, location (and in practice, non-manual markers, but that’s a whole conversation in its own right), and due to the physical nature of them, they all happen at the same time (save for dynamic signs, but for the sake of simplicity, we’ll ignore that here). That is to say, I couldn’t do a handshape, then the movement, then the location, then the palm orientation - that’s not how bodily mechanics work. Theoretically, someone learning a sign could go “okay, it has an A handshape, palm orientation is facing across the body, moving up and down, located in neutral space making contact with the other palm which is facing upwards with a B handshape.” In practice, doing that much examination is more complicated than seeing a sign and copying what they do, though some element of that could be included in making adjustments during learning.
As previous respondents mentioned, ASL and other sign languages don’t have a standard writing system - there have been a few systems created with that intent, but never really took off save for small groups of people here and there, and those are often teachers or linguists. The vast majority of signers don’t know those systems, and many have never even heard of them.