Literally hours have passed (for Brand and Cooper) between falling into the black hole and Cooper waking up on the Cooper station: remember, all the oxygen he had after ejecting from the shuttle in the black hole was the oxygen in his space suit. He was found with only few minutes of oxygen left (as the doctor on the station told him).
So - when he woke up on Cooper station, Amelia was still on her way to Edmunds planet. Murph was expected to arrive on Cooper station "in a few weeks", so that bit of time has passed between waking up and the talk between Cooper and Murph. Even though Edmunds planet was "farer away in the field" (farer away from Gargantua), those few weeks were enough for Brand to reach the planet and to start setting up the camp. So far for the time line of Brand and Cooper.
Now how about Earth's time line, what happened there in all those app. 51 years, why does Brand not arrive on an already colonised Edmunds planet? I also believe the main priority after solving the equation was to get the people to safety, which does not necessarily include risky further expeditions when you have no clue, no confirmation whatsoever if there really is a habitable world on the other side. No one came back from the first expeditions before Cooper did. They have a comfortable life on their space station habitats now and sending an entire station through the worm hole is out of the question, I'd say - you would risk too many lives if you'd send send so many people at once on a station, i.e. on a huge vehicle, it would be exposed to high forces (remember even for the Endurance it was a rough trip). No one would risk something like that, I think. So we have to think smaller - smaller expeditions, on small vehicles with a couple of further explorers on it: why were none sent in 51 years?
It will be soon 50 years since humans' feet first touched the moon's surface. Why weren't "we" able to land on Mars even once in all those years?
My attempt to find an answer (for the movie now of course) would be rather simple: Priorites have changed. People safe on their stations, having normal lives, their children attending universities again, and still in their own solar system, nearby the planet that they can try (at least try, at least help) to "repair" after evacuation. A totally destroyed biosphere is nothing you can just repair in a couple of years nor decades nor even hundreds of years but humans do know their world is a jewel: sure, we're a throwaway society but even the film does not try to deny that, remember Coopers words when looking back at the spinning earth slowly getting smaller in the window on the Endurance.
That does not lower the yearning to go the next steps into space - into interstellar travel, but they needed a cause for that. Just imagine the mars rover would have found a microbe there, do you have an idea how quick suddenly NASA would have gotten the ressources from the gov to get a first crew there?
They needed a cause. Before Cooper was found near Saturn they had none. There was silence from the first missions, while their urgent needs were already met (equation solved, space stations working). Only after Cooper was found, they learned from him how their mission worked out, learned about Manns and Millers planets, and learned about Edmunds. We can absolutely expect that this has sparked the preparing of next missions ("our new home").
I would however not expect them to be space stations being sent through a worm hole, but Endurance-sized ships as backups for Brand. And even if Cooper was obviously not willing to wait that preparation time out, we must not think about having them to stay there alone for long.
(That being said, 51 years is still a lot for a society that just found the solution for gravity to not send further expeditions through a worm hole, agreed... but then again, almost 50 years since touchdown on moon, remember.)
But when thinking a bit further, an other possible reason shows up: what if there were next missions? Imagine an Endurance 2, with 3-4 people on board, a comparable mission sent as a repeat mission after they heard nothing back from the Endurance: imagine this next ship arriving at the other side of the worm hole. Now what. There will still be a very similar situation: Millers planet still "coming up fast", sending very promising data, still echoing its inistial status report endlessly. Manns planet will now be as silent as Edmunds will still be (as Brand will not yet be there, still making her slingshot around Gargantua with Cooper). Those both planets (Manns and Edmunds) could have suffered a transmitter failure. Or not. While the nearer Millers planet will be sending its echoes. So what will this crew do? The next mission after Endurance might be already on the other side of the worm hole, however still busy with orbiting Millers planet (and experiencing the time shift that this orbiting would cost), preparing to land there or having aready landed and being either already dead or - if survived for a couple of hours - on their way back to their orbiting ship, all caught in the time shift on Millers. If they had the same bad luck as Miller, how about the 2nd, 3rd expedition? All arriving on the other side and being faced with the same situation: a nearby planet too close to Gargantua but with "very promising" data, "still transmitting". While 2 other planets went silent. What I mean is: the fact that Brand arrives on an uncolonised planet does not necessarily mean there were no other expeditions at all, it can also mean they were either not successful or also experienced time shifts.