Hello, and welcome. I am a licensed attorney and happy to assist
Theoretically, unless the initial contract provided that the date could be extended at the discretion of the gym, its doing so is questionable, at best. It could be argued that extending the deadline effectively dilutes the pool of applicants, in which you have established yourself as a significant contender, and on that basis you sustained harm. On the other hand, the gym could argue that an extension gives you additional time to bring in more members, so the extension caused you no prejudice. That said, if you could show a court that your potential pool of applicants was limited to those actually obtained, or that you were somehow limited in your personal or professional life by the time constraints set forth in the initial contract, you would likely win if the issue was taken to court in the form of an action for declaratory relief and/or an action for injunctive relief
However, bringing a court action would take significant time and energy, and financial resources if you decided to hire an attorney, so it's important to weigh the pros and cons before determining how you should move forward. In any event, the law is on your side, and perhaps presenting that fact to the gym would be enough to convince the powers that be to honor the original terms.
I hope this information helps.