Do those employees deserve a living wage?

No context needed, unequivocally, yes.

It seems to me to be equally plain that no business which depends for existence on paying less than living wages to its workers has any right to continue in this country. - FDR

It seems that where we differ is they yes, I do think that employee providing shitty service deserves a living wage. But more importantly, there's so much that people do for us that should be paid but isn't, or that isn't paid enough to live off of but should be. People who volunteer their time to clean up public spaces, to help take care of the sick and elderly or young children, people producing art for the rest of us to enjoy, people doing doing fundamental research on topics that aren't currently trendy, and likely many more that aren't coming to mind right now.

Tell that to folks having to work three jobs to provide for their family. The concept of the minimum wage was to provide enough wages for people to support their families (on a single income no less) with a forty hour work week. Now, there isn't a single state where you can afford an average apartment on a minimum wage job. Saying 'youre paid what you're worth' is just a fancy way of saying some people don't deserve essentials in a post scarcity society.

But yeah, we can just let kids work to add a third income to the household. That'll solve it.

Have you not been to a restaurant where the service was terrible? Do those employees deserve a living wage?

I've been a waiter and can tell you it's probably because the boss won't pay for enough people or is trying to cram too many people into their establishment without investing in the required infrastructure and staff.