At first it seemed to me that the cover of that model is there mainly for visual purpose, as its diameter is smaller than the outer chainring.
But then I discovered that your model, on the plastic guard also has a plastic pin in between the crank arm and the chainring. Its purpose is to prevent the chain from doing damage to the crank arm or chainring should it slip over the big ring. In case the limit screw is not adjusted properly, for example. So it has a safety purpose, other than visual aspect. Removing the plastic cover would also remove this safety pin.
I have to disagree that chain covers are there mainly to protect your clothing. They have other purposes too. Like the pin your model has.
Or this model, for example, that has a bigger guard, and it protects the rider in case of a crash from the sharp teeth of the chainring, which can become "ninja star" as some people describe them:
Regarding the creaking noise.
I would inspect the crank arms for cracks too. And the chainring fastners for looseness and play. I believe yours are rivetted. The bottom bracket can cause creaking noises as well, it's the usual suspect.
In reply to DoNuT's comment.
Fair point. I agree with you, I've seen models where the pin is part of the crank. But this one seems to have the plastic pin as part of the cover. It's cheaper to make it like that I guess. Here is a picture with the cover completely removed:
Note the pictures in my answer are from a Tourney Triple, OP has a Tourney Double, but I suspect this plastic cover is the same.