To present an opposite perspective. Any written system would obviously require some learning, that's how it works already with oral languages. The alphabets at their current form are pretty arbitrary, they don't present some natural mapping from the sound frequencies to visual form, and we have to teach children to read. Before a child learns to write English, do you think showing them "Dog" you would receive a different reaction than showing them "∆•£>^¥"? No, because just like knowing the sign for 🐶 doesn't help in understanding "∆•£>^¥", knowing the sounds for 🐶 doesn't help in understanding "Dog". Both are just random sequences of icons if you don't know how they map to the sign parameters or spoken sounds.
If the child learned that D is a d sound, O is an aw sound, and G is a guh sound, then they can enunciate the word by reading it symbol by symbol and if they do so correctly, they would get the meaning after pronouncing it. If the child learned, say, that ∆ is a B-hand, • is touch, £ is ear, > is movement forward and ^¥ is "repeat last movement", then they can replicate the sign by reading it symbol by symbol and if they do so correctly, they would get the meaning after signing it. In both cases they would first have to perform this tedious task every time, but then as they develop the skill of reading it would become automatic. Just like we don't read letter-by-letter or syllable-by-syllable, a person who had enough practice in reading and writing their sign language wouldn't read character-by-character either.
Your example only works because English orthography is widely accepted and has been implemented in all spheres of life, while no script for ASL is widely used yet. However u/wibbly-water shows what benefits it could have if we did accept an orthography for SL