Generally, this is a nicely written opening paragraph but it lacks a clarity of purpose. You are presenting (at least) three ideas that do not all coincide with one another.
- The American Dream is an illusion.
- The American Dream is not an illusion in itself but is often distorted into something "other" which is impossible to achieve.
- Only those who are born wealthy can see the flaws of the American Dream.
You should pick one point of the first two to use in your argument. Using both is not going to lead to a logical or strong argument.
The third point here is highly questionable as it relates to the novel. Though this is an interesting point, it will be difficult to prove. Remember, Jay Gatsby rose from modest means to great wealth. Despite the fact that he failed to recognize the failures of that wealth to help him integrate into "high society", he was in the perfect position to make this recognition. He did not have to be born into wealth to discover how the fulfillment of the American Dream does not lead directly to sophistication, social acceptance, etc.
The question posed by your teacher is a great one to follow in re-shaping a thesis. You propose that people cling to a distorted version of the American Dream. Defining this distortion as it exists in the text will give you plenty to write about.
Figures like Nick and Gatsby seek entry into a world that is supposed to be sophisticated and even erudite, but it turns out to be a shallow and tawdry social echo-chamber, wildly singing its own praises and celebrating its own dramas - a far cry from the hallowed image of an American success of quiet, stoic, and dignified wealth.
A small point: The term "deliberated" should be replaced by something like "commented", "ruminated", or "reflected".