Notes with a number on it isnt a confident or mature tactic to me. Why not let her say no through text rather than saying no through literally throwing the note away and thinking nothing of it? it also lessens the *anticipates text* phase. When someone texts you the simple obligation of replying back is a pretty sincere gesture. It isnt a gun to her head
You know what it doesnt matter what we think dude should just ask her out however he feela comfortable
Asking something out directly, while they're working, really puts them on the spot. They can't afford to be rude, because "the customer is always right" and a terrifying amount of managers are shitty people that will yell at their employees for any little reason (and the non shitty ones usually get out of Dodge at first opportunity because it's shitty). You don't think an employee can get in trouble for turning down a customer's advances? Man, I've seen people get in trouble for that, and I've seen the employee be nothing but polite.
Asking them out at work directly forces them to make a split second decision about you, a customer that is really just making their life more difficult at this moment (thought process: oh there are orders to fill and my coworkers are snickering and god my feet hurt and aaaaaaagh). Handing a note give her time to think, doesn't interrupt her work flow, and is more discrete than going up to the counter and saying "HEY YOU'RE CUTE WANNA GO ON A DATE?".
Asking someone out at work in *any* fashion is not ideal, but a discrete way is far better in this situation than a direct one.
EDIT: Wait, did I miss something Max? Are you suggesting that Lucky try and ask for her number instead? That sounds legit harder than just bluntly asking for a date to be honest.