I think it would

It may break some clients. Don't try creating your custom thing, just follow the spec https://lightningaddress.com/

That's a good question. I'm not sure if that would break it.

Does the "$" really makes sense in a LN Address context?

I kinda understand it in a "$username" as a brand signal. But in a LN Address that brand is on the @cash.app.

It looks confusing, imho.