Should the QA team be doing unit test?

I come from a configuration management background using tools like chef. I have done quite a bit of code testing. Recently, I have been entrusted with the responsibility to work on the CI-CD of the various applications. But I am noticing a culture that’s quite not in line with my ideology.

Let’s say we have 4 environments in the CI-CD pipeline: Dev, Test, Stage & Prod. The dev environment is used by the developers to deploy and test the app before rolling out to the next stage (test). The test env is for the QA team to run their test. And STAGE is another env for 2nd layer of test by the QA folks before the code goes to prod.

Now, does it make sense for the QA or the CI process to run unit test while progressing/after deploying the code to (TEST or STAGE)? I agree that the unit test should be a part of the automated build. But unit test is mainly for the developers, for their code testing. The QA should be focus on the functional testing, load testing etc.. using their frameworks which may be selenium/protractor or LISA. Why should they be focusing on junit or nunits?