I think your assumptions are a little flawed. "The Great Wall" as we think of it (the large stone structure) only really came into that form during the Ming Dynasty, following the Mongol invasions and the Mongolian Yuan dynasty. Prior to this, there was a series of rammed earth (as most historic Chinese structures were made of this, where you have a wooden frame filled with densely packed earth) walls dating back quite a ways. So there's the first problem with the assumption. It was a combination of defensive power and a showcase of Ming imperial power. The only major breach of note after this time was at the end of the Ming dynasty, when rebels held Beijing and rebellions were cropping up throughout the empire. A general named Wu SanGui who was in charge of guarding the famous Shanhai Guan (山海关, the pass in the wall where the wall meets the sea on the eastern end) actually allowed the Manchus to pass through and joined them to retake the capital and form the Qing dynasty (that's a whole other story).

As for your other part about "wiping out" the Mongolians, consider: the Chinese Empires were quite large (China today is quite large as well). This means there was a lot of border to patrol in the first place (you have massive efforts starting in the Tang Dynasty to guard the western borders, and of course the aforementioned walls in the north as well as the large forces along the "Great Wall"). These were aimed at defending against big attacks (say, Tibet) and limiting the raiding of steppe-nomadic/semi-nomadic (mostly Turkic) tribes. At times where empire strength was low, these tribes would raid imperial land in force and thus interfere with tax collection (a farmer can't pay his full grain tax if some raiders came and took a large portion of the harvest). At times when the empire was strong, these interactions between the imperial lands and the nomads were more along the lines of minor raiding and (best case scenario) trading. Why not take care of this problem by "wiping them out"? For one, these nomads were useful. The same people that raided Chinese land were often employed in Chinese armies (sometimes acting in large groups of auxiliaries in emergency times, in exchange for raiding/looting rights on some captured city). And the other problem was, again the emphasis on "semi-nomadic". For Mongolian land specifically, there's quite a bit of it, and these tribes are spread out over the harsh terrain. It's a tough life. It's even harder to conceive of some large Imperial expeditionary force trekking all around this land, constantly chasing after these tribes when it's a much better use just to have these troops defend specific points (and less expensive. And more flexible if they're already within Chinese borders).

Edit: I spent a little bit of time talking about the whole "semi-nomadic" thing, but so what? Well, I wanted to add that besides moving around a bunch and having no real sort of capital to take, there's also a power structure in play that makes it tough to really "eradicate" the Mongolians and other Turkic tribes. The invasions by the Mongolians under Chinghis and the Yuan dynasty started by Kublai were times of extreme organization and centralization. But even under Kublai, you don't so much have a "Mongolian government/organization" of the Yuan dynasty so much as you have a "Mongolian-staffed/led Chinese dynasty". Same thing happens with the Khitan and Jurchen prior to the Mongolians coming around: the only sort of organizational comparison with something like a Chinese dynasty would be for them ruling Chinese land (someone more familiar with the actual Mongol, Jurchen, and Khitan tribal politics when not ruling over vast amounts of land in a central system would be welcome to fill in some blanks here). So as I said, these groups of people as raiders is inevitable and there's no real way to "hit that at the source", and really that's more effort than it's worth (the Chinese viewed it this way as well. Unless someone was causing a big problem, like if a local Khan was creeping too close to Chinese territory and/or hitting too hard in raids, it's better just to monitor and defend against rather than go at an all out war). Same thing happens with the Manchus: you have groups of Mongolians, Khitan, Jurchen, and other northern Turkic peoples that individually as raiders are not a huge threat (this is after the Yuan), but once they unify under the Manchu banner they obviously become a bigger problem.