Ordinarily, the cover does nothing but hold the brake fluid from spilling. However, there is one important thing you can easily check.
As the pads wear out, the pistons are being pushed more out, and the level of fluid in reservoir goes slightly down - which creates a vacuum in reservoir. If the vacuum in resevoir stays, it's like someone pulls the pistons back. So, when you squeeze the lever, the pistons need to go an extra distance before they engage. This creates a preception of mushy levers.
As Tufbusa pointed out a while back, to combat the vacuum, every stock cover has two small notches on the edges of the cover where they contact the reservoir - take a close look at the stock cover. If your aftermarked cover does not have those, or they are not cut properly to relieve the pressure or vaccum, this may become a real safety hazard.
If they do not compensate for the vacuum inside the reservoir, the brake feel mushy. However, if the pressure builds up inside the reservoir (e.g. on a hot day, or during aggressive braking), it would act in the same way as if you are pressing on the lever. This may cause a lock up of the front brakes with daring results.
So, if it turns out your aftermarked cover does not have ventilation slots, you can either make them yourself (similar to the stock cover), or demand replacement from the seller.
Let us know your findings.
Also, to verify this theory, you can simply open the brake reservoir cover (this would equalized the pressure), and see if the brake lever feel gotten better right away. You would have to pump it a few times though to move the pistons back to being as close to the pads as possible.