The thing about time travel stories is that you can usually reconcile them given enough thought and energy, but in doing so you will have put more effort into logical consistency than the actual writers of the screenplay did. So in noble time travel tradition, here is my attempt at explaining the time loops in the movie.
The first thing to realize is that there are two time loops. There is the time loop where Coop, the main character played by Matthew McConaughey, sends a message back to himself and his daughter, Murph. Another “outer” time loop may exist because future humanity sends a wormhole back in time and allows Coop to send a message back in time via the singularity at the center of the black hole.
Secondly, you have to enter a time loop from a reality where the time loop does not exist. That is where this analysis will begin, in the Alpha timeline. We know nothing about this timeline except that it eventually results in Coop and Murph receiving a message from the future and that it has a wormhole and a time radio in the center of a black hole. If we take the movie’s word, these things never occur naturally and either a future humanity or aliens placed them there. If it is a future humanity, this causes an issue as it requires humanity to survive WITHOUT the interventions in time played out by Coop in the movie. It will eventually gain mastery over time and, despite surviving the destruction of Earth, they must decide for some reason to tamper with their species’ past and send some sort of message or wormhole or something. This starts the time loop. Alternatively, there are in fact aliens and they come across the artifacts of humanity and try to save us by reaching into the past. Or maybe aliens created it in the present and there is no time travel to set up the movie. As viewers we will never know what the Alpha timeline is, but let us assume the movie is correct that future humanity created the wormhole and the time radio.
What we do know is that eventually someone sends a message to Coop and his daughter in her room. Maybe future humanity does it directly or, more likely, in the timeline previous to that in the movie, present humanity discovers the wormhole and the time radio in the center of the black hole and send a message to Coop and Murph. Maybe that was Coop, again we will never know.
What about this STAY message they receive? Ostensibly, the only person that would send such a thing is Coop himself, but he can’t send that message unless he has already gone to the black hole. This is where it gets really tricky, because every message that changes the past creates its own time loop and the rules of time loops are like programming loops, each loop must be completely nested in the outer loops. So in fact, there are more than two time loops, one for every message that Coop sends into the past. Lets see if the movie accomplishes keeping them straight.
To backtrack a little bit, someone, maybe Coop himself but it could be anyone, sends the coordinates of NASA to Coop backward in time. Then Coop goes on the mission and reaches the center of the black hole. When he gets there, according to the movie he first sends the message STAY to a point in time AFTER he received the NASA coordinates. This is a problem. Because you immediately spun off a new timeline without anyone sending Coop the coordinates to NASA so he will never leave Earth after seeing the coordinates to send the message STAY. In order for this to work he should first send the coordinates in order to ensure continuity with the current timeline and his arrival at the black hole. At that point he can alter the timeline.
You can of course posit that the singularity is special when it comes to time travel, but the necessity of Coop’s presence to send a message to his daughter was already tenuous; it’s even worse if we assume that future humanity has such enormous time traveling powers that they do not create a typical time loop. Is there some other way to reconcile this? Perhaps. If in another timeline someone sent the coordinates first and then said STAY all would be fine when Coop shows up and says STAY first. The problem here is that Coop is not thinking about the rules of time travel and to think he said STAY at the same exact time as needed to ensure congruity with his past is a bit far-fetched. To reiterate, someone, not necessarily Coop, familiar with time travel had the good sense to first send the coordinates and then attempt to get Coop to stay with a message, maybe Coop went crazy and this person wanted to stop him from coming or maybe this Coop was just better at time travel. Coop then gets to the center of the black hole and ensures congruity by accidentally saying STAY at the right time.
However, a more plausible scenario is the following: despite what we perceive, the first message was in fact STAY. That is, Coop or someone else, in a timeline where no messages from the future have been received in Murph’s bedroom, makes it to the black hole and their first message is STAY. It is convenient if it is Coop because we can expect his emotional response when encountering the time radio to always be to try to get himself to remain at home with the STAY message and thereby unintentionally ensure congruity with his past even when he is not thinking straight. However, this causes another problem. Why would Coop need the coordinates of NASA if he manages to get to the center of the black hole without them? We must then posit that Coop saw some exigency that he thought could be avoided if he has the NASA coordinates. Maybe his arrival at NASA earlier elides a catastrophe. Anyway, he sends the coordinates and all future Coops realize, after their initial attempt to stop themselves from leaving Murph, that they too must send the coordinates or risk a paradox.
With those two loops out of the way, Coop could then implausibly transmit data via an old watch to his daughter to create one more timeline. The watch avoids all these time loop problems because the message is sent last and it arrives after Coop has already left the planet and thus cannot affect him sending it.
From here we back out to the possible future humanity time loop. This future humanity appear happy with the trajectory of humanity, but knowing the rules of time travel, they must make sure to send a wormhole back and create a time radio for Coop to send his message or risk creating a paradox where they never come into being. Or maybe they are not happy and the timeline we see is just one timeline on the way towards a stable timeline. For that matter, future humanity might be stuck in someone else’s time loop, including an even more future humanity.
To conclude, I summarize the most plausible sequence of events with the assumptions of the movie that this is an attempt by future humanity to save present humanity. Humanity survives the apocalypse without help from the future and gains the ability to tamper with the past and does so. An indefinite number of timelines proceed without any paradoxes erupting that leads to the first timeline we can speculate about where present Earth discovers a wormhole to a time radio and then someone, very likely Coop, sends the STAY message to Coop and Murph in her bedroom. This branches another timeline off where Coop again goes to the time radio and sends STAY but then realize he can avert some disaster by telling himself the coordinates to NASA. This branches another timeline where Coop uses the time radio to send three messages; STAY, the coordinates and the observational data. This creates another timeline, the one we see in the movie. This averts the destruction of the human race. The new future humanity remembers Coop and Murph and realizes that in order to avoid a paradox they must send a wormhole and a time radio back in time for them to use. At this point they could leave the past well enough alone or try to alter it again to be different than the events we saw in the movie.
This also provides a much better interpretation of the movie than that Coop and Murph are special and that future humanity needs Coop’s love of Murph to save humanity. Instead, what we know is that future humanity creates a wormhole and a time radio. They do this because they don’t know enough about the past to change it and instead give present humanity the tools to save itself. Once present humanity starts creating time loops, future humanity is essentially locked out until they stop doing so. Coop’s interpretation of events relayed in the black hole is then wrong, it was just chance that brought him there and the fact that all he sees is Murph’s bedroom is an extension of his own mind. The rules of time radios inside of black holes are of course unknown, especially to Coop. In this way we can ignore the terrible love motif.