Doubtful the shifter doesn't work right. I've assembled thousands of bikes with cheap low end shifters and if they were shimano shifters I never had one that failed to work. It can happen but it would be incredibly low on my list. Often if a shifter was replaced and the shifting was crap or a new build with the same problem, it was due to a small kink in the cable typically up near the shifter. They love to kink just inside the barrel of the shifter and while it might look like it should be nothing the internals of a 4mm housing is tight enough it can matter. Pull the cable completely out of the housing and look for a very slight bend, that's probably your culprit. Often it allowed for decent but not great upshifting and would feel like the shifter was a little tight to push but was more evident on the down shift with slow downshifting and noise.