When people talk about "What is Taiji skill?", I always like to ask, "How to develop/test those Taiji skill".
I think it's interested to discuss "How to use Taiji skill to deal with clinch". If such skill does exist, all the wrestlers in the world will come and try to learn that skill.
There are certainly ways for Taijiquan to deal with the clinch - both in a civilized way and the unsportsmanlike way. The reason I bring this up is because like many Chinese martial arts, targeting the throat and other areas is within the old-school version of the art which is not nice or merciful to the opponent.
To go off on a tangent, one of the differences between "old-timers" in Chinese martial arts and modern competition Push Hands (as well as the sport of Shuai Jiao) is that strategically, they do not grab the opponent's torso with both hands.
So here is a martial art from the Zhang Family, and he talks about how Shuai Jiao has a sport mindset where they like to grab the waist and don't have hitting. But the opponent is offering up their eyes, throat, nose, etc... to you - all while both of their arms are preoccupied.
树立短打理念 是在优酷播出的体育高清视频,于2012-01-28 12:18:22上线。视频内容简介:树立短打理念
v.youku.com
So when it comes to modern Push Hands, they have a sport mindset and a lot of their vulnerabilities are protected by the rules such as forbidding head control.
But I also don't want this to be an excuse or a replacement for more "civilized" and skillful approaches to dealing with clinches or being grabbed around the waist. Those exists too; but you probably can't find them on the internet as it relates to Taijiquan.
But on a strategic level, the mindset between old-school Taijiquan and modern competitive Push Hands is very different.
If you keep sweeping your opponent's foot, your opponent has to deal with your foot sweeping before he can do anything. It's so important to train how to deal with foot sweeping. But both foot sweeping and counter to foot sweeping are missing in Taiji PH training.
Sweeping and counters to foot sweep is all over the place in the PH I train. I love sweeping and tripping. Many applications combines sweeping and throwing together.
This is a matter of content being lost over time. Not all styles of Taijiquan have certain things. Many things are just lost to time but survived in other branches. This is also where people will get upset when you point out that they don't have X, Y, and Z.