What does "promote hard work" in this case mean?
Does it mean they should promote people who work hard too?

-----------------------

Boss: I appreciate all the perspectives and feedback you provided to me. I wanted to make sure that your voice is heard and the voices are heard of all your team members around you.

Worker: I think every worker wants to feel appreciated for what they do. We want a system that promotes hard work.

Question

What does "promote hard work" in this case mean?
Does it mean they should promote people who work hard too?

-----------------------

Boss: I appreciate all the perspectives and feedback you provided to me.  I wanted to make sure that your voice is heard and the voices are heard of all your team members around you.

Worker: I think every worker wants to feel appreciated for what they do.  We want a system that promotes hard work.

Answer

It means rewards, encourages hard work. Not necessarily actual promotion.

Answer

I agree with GuideDogSaint. Promote as in encourage, incentivise.

Answer

Honestly that doesn't sound like it was written by a native English speaker. But perhaps it is more British so I've never heard that in North America.

I would think your interpretation is possible, but would have expected something like "rewards hard work" as the phrase, at least here where I am.

The other way I might interpret "promotes hard work" would be the worker is saying the company needs to have policies and incentives, and perhaps supervision, that ensure everyone is working hard and some don't get to be lazy with other's having to pick up the slack.