Would I be the asshole if I suggested a kid with a stutter learned sign language?

Would I be the asshole if I suggested a kid with a stutter learned sign language? Hey all, thanks for reading. I recently met a kid (teenager to be exact) who has an extremely bad stutter. He was alone at a gaming cafe having a convo with the owner, using his phone to type. I asked him if he wanted to play something together and he enthusiastically joined. He's a very polite kid, and constantly paused the game, got his phone and typed "thank you" with a smiley face every time he was appreciative of something. A while into playing he typed on this phone apologising for not talking, which he said was because of his stutter. I said it was fine and encouraged him to talk if he wanted to and that I didn't mind waiting for what he had to say. After that he managed a few words but still had a lot of trouble wasn't really able to finish a full sentence. He teared up while typing to me saying that he felt really ashamed that he was forcing me to read instead of being able to talk. I don't know anyone else who has a stutter, but I can tell he has a pretty bad one and his extreme anxiety makes it a vicious circle. Here's my question: would offering to teach him a little sign language be a boost to his confidence or would it be a hindrance to his improvement? Maybe just some basic words like thank you so that he can feel like he's just playing a game uninterrupted like a normal kid? Another thing is if its potentially offensive, I would hate for him to think that I'm impatient or I'm teaching him just so I don't have to deal with his stutter. I don't really know anything about stammering and Google wasn't too helpful, so thank you for giving your perspective on this! bonus question: if you're taking a long time to say a word (he'll take up to half a minute at times), what is your preferred response? like where should I look? and if it's because of anxiety do you want some reassurance or encouragement? etc.