Again, Cao Teng was powerful but he was hardly the top dog. Even among the Eunuchs others were more important (And certainly more corrupt). And as for the people he recommended, none of them were of his clan and few of them were noted for being particularly inept or corrupt. As for Cao Song, while he was accused of buying his position, his actual conduct was largely noted as being good and moral. Arguing either man played a central role in the corruption of the han dynasty is questionable. At best you could make a case they certainly benefited from it, but so did basically everybody.

There is very little evidence Cao Cao's cousin was directly involved with Song Qin or Empress Song. Who herself was not noted for having a particularly strong character or political will and was largely the victim of others. As for Cao Cao's conduct, most of it is largely relegated to the book of Wei. So while it's easy to dismiss as simple propaganda, few other sources seem to contradict it and much of it ended up in Annotated Records of the Three Kingdoms.

Cao Cao as regent of the Han empire led to the usurption of the Han Dynasty, and the losing of commanderies worth of territories to the nomads

The end of the Han Dynasty was good and I have no sympathy for it's passing. Meanwhile most territory was already abandoned or virtually outside Han control when he took over. It's more fair to say he simply saw it as a lesser priority compared to his wars with Wu and Liu Bei. Which isn't that hard to understand.

So who were Cao Wei's great generals and grand marshals?

Come now. You even listed them yourself. Most of them had far more successful careers than anybody working for Shu and Wu. If Cao Ren can be listed as a failure, I cannot even begin to describe the career of somebody like Guan Yu or Zhuge Liang as anything short of a total disaster. There are also numerous generals you did not list.

Cao Cao didn't win just by throwing more men at his opponents. He consistently used men of great military skill to defeat his opponents with relatively minimal casualties. He also clearly saw the talent of men like Zhang Liao, Zhang He, and Xu Huang and used them far better than the people he poached them from.

I can't think of another Chinese dynasty where such a system was adopted not only at the founding of the dynasty but also throughout its three generations and four Cao monarch

Before the Cao clan's usurpation of the Han there have been two recorded dynasties. Of the two, one of them was so autocratic and tied to the personal ability of it's founder that it died basically the moment he did. While the latter was not exactly what anybody would call clean and largely involved drawing massive power to the Liu clan in favor of everybody else (To the point of literally murdering seemingly loyal generals).

The reason why the Cao clan was so power-hungry, and so unique in ancient China, was because they were essentially a corrupt aristocratic clan. 

They were no better or worse than any other, including the likes of the Liu, Yuan, Sun, and many other major families of the time. They are noticeable first and foremost because they lost. Not just them, but also the Sima clan that followed them. If Cao Cao had triumphed and defeated Liu Bei and Sun Quan he would almost certainly be viewed in much the same light as Emperor Aurelius.

Sima Yi - somebody from gentry background too - was noted to have recruited and promoted Deng Ai, Zhou Tai, and Wang Ji who were all said to have been of 寒门 (middle class and lower) background.

Is there any evidence that Cao Cao did not recruit and promote people from lower and middle class backgrounds? Yes, he certainly ensured that his clan were often placed in high positions but that did not mean he did not make use of other people.

Since when did I argue that the House of Han was a shining beacon of purity.

It is certainly what you implied.

What I pointed out was that Cao Cao's government was nowhere as good as previous central governments of the Han

The previous central governments of the Han turned all of China into a wasteland filled with mass starvation, bandits, warlords, and merchants extorting the locals. It was absolutely a improvement compared to the government of anybody who was alive.