Dear Simple,
If the point of the clothing is to cause the wearer to identify with something, than the clothing itself has no meaning. It is causation of a purpose. Placing the clothing on a mannequin would not indicate what identity it portrays. Only when it is worn by someone involved in an activity does it take on meaning. (Baseball cap and Bigdei kehunah.)
Standard dress, is not part of any activity. People who dress a certain way (in a traditional sense) will always wear their outfit, regardless of what the activity is. One wears when they go about their routine, in much the same way all people wear clothing. I am typing and you are typing. That we are dressed differently, does not mean that we are typing differently. Or that my typing serves an overall purpose. The identity would be in the clothing itself. But the above paragraph demonstrated that even clothing designed for a specific task do not have an intrinsic meaning.
Here is how I remember the shtickel from yesterday. The bigdei kehunah identify as the service of Hashem because of their association with the service. That is to say the service itself would not externally demonstrate the purpose without clothing. And why a mets cap is silly because baseball is an activity without an overall purpose, so there is nothing to demonstrate. Yet, when it comes to wearing a black hat or whatever else, it could have such an effect on the wearer that he should be proud that he announces what he identifies with. Where would this effect come from? In the essay, the point is made that people go about everyday life without reflecting on what their actions mean. So it is not the actions or the clothing. It is the mental effort. And that goes into tzitzis. Which is there to affect our considerations. But that is specifically given for that meaning, which makes it an intrinsic part of the clothing.
Here is the contradiction.
On bigdei kehunah. It starts out that the greatness is because of the activity, and what that signifies to the onlooker. But at the end it should be the same as tzitzis, and the meaning is intrinsic to the clothing and intended for the wearer.
On baseball caps. The essay says they are silly because it is just a game. But if any hat makes the thinking wearer proud, than the wearer of the mets cap would be proud when they win. And, the makers of the cap want people to think more about the mets, so it is a similar idea to the paragraph above.
On traditional dress. It would mean something if people would think about it. But it is demonstrated that they do not. That is what tzitzis is for.
If anything, Rav Miller ZT”L is concluding that people will identify and be proud of what they wear. Even though clothing itself is not about ‘identifying’ or being proud.