Hmmm... I think the answer to that question is "it depends."
If you are not rinsing the mechanical filter media weekly, then you are probably better off without it. But if you rinse it weekly, it shouldn't be adding to a nitrate problem. In other words, if you're doing good preventative maintenance to start with, the media isn't hurting you.
Bioballs are really good at turning ammonia/nitrite into nitrates. It's their job. If you have a fair amount of live rock, the bioballs are probably not doing much for you so you could get rid of those. But if it's your only biological media, they need to stay.
A protein skimmer is good, but is not going to be the magic pill for nitrates if you are overstocked, are overfeeding, don't do water changes often, etc.
And last but not least, once you have the nitrates, the easiest way to remove them is with a water change. Once the nitrates are in the water, a protein skimmer isn't going to take them out.
You can have very little nitrates, have no protein skimmer, and have mechanical filtration media all at the same time if you're up on your maintenance and husbandry. It just really all depends...