In Interstellar, why does Cooper leave to go find Brand at the end of the movie? There's no wormhole for him to travel through. Without the wormhole he can't get to her, otherwise someone would've done it before him.
I see multiple questions here.
I'll try to address each one:
Why does Cooper leave to go find Brand at the end of the movie?
The best explanation for this is that the only thing Cooper has left in life is Brand. Through the entirety of the movie his primary drive is to return to Earth to be with his daughter. When he is finally able to return she is on her death-bed, surrounded by her own family. She specifically tells him that there is nothing left for him there, and that Brand is waiting for him elsewhere.
There is no wormhole to travel though:
It's a bit unclear if the wormhole is still present or not once Cooper returns to our solar system. What is clear is that the formula he sent back to his daughter while in the tesseract was enough for her to learn how to manipulate gravity, thus allowing the creation of the facility he arrives at, and for the possibility of creation of new wormholes.
Without a wormhole he can't get to her, otherwise someone else would have done it before him:
Again this is a bit unclear. It is possible other people managed to arrive on Edmund's planet before Cooper, but it is never explicitly shown. As for the wormhole to get there, see point 2: The wormhole might still be present, and if it's not humans now possess the technology to manipulate gravity and create new wormholes.
Cooper specifically leaves to go and find Brand and this must be via the wormhole.
The destination is clear from his conversation with Murph at the end:
No parent should have to watch their own child die. I have my kids here for me now. You go.
Brand. She's out there setting up camp. Alone in a strange galaxy. Maybe right now she's settling in for the long nap by the light of our new sun in our new home.
As for your statement which forms the main part of this question:
There's no wormhole for him to travel through. Without the wormhole he can't get to her, otherwise someone would've done it before him.
I understand why you think this, but he has to be going to find Brand. This is because:
- This is Murph's suggestion. Murph wouldn't suggest this if it were not possible.
- There is no mention of the disappearance of the wormhole. Its been used several times already, by the 12 Lazarus missions and the Endeavor at least
- Because of time-dilation very little time indeed may have passed on Edmund's planet. Perhaps other missions have already gone out to find her (and the other Lazarus mission crews). However in her case it's possible that they are still traveling there. Much time has passed on Earth, but still very little time has passed for Brand and Cooper.
So, despite other missions that may have left for Edmunds planet, there is no reason why he shouldn't follow.