I get where you're coming from, but I don't have a problem buying it. There are two ways of explaining that, as I mentioned. If nobody told them who was coming (remember Murph is like the fucking queen, so when the space station people found Coop they told her and then she could've told them not to tell anyone else) then it's easy to explain. This is my interpretation. Remember how nonplussed the doctor and nurse were with him when he woke up. The doctor treated him like a nobody and the nurse laughed when he had the gall to think that "Cooper Station" was named after him. They found an astronaut floating in space. They might not even know/believe the time travel/wormhole shit. They might think what everyone else thinks: That Murph figured everything out herself and saved the world and that her crazy shit about her dad was just in her head and the "inspiration" she used to figure things out. With everyone, including Murph's family,
she is the important one. Coop is a memory and a name from long ago if that. Hence the line from Coop's wife, that the function of parents are to be memories for their kids. Coop is a memory, a ghost. I'm telling you, it works. But even if you don't buy this, even if you want to insist that the family knew who he was and what he did, I'd still buy them showing way more affection toward Murph and cautious/confused/respectful distance toward Coop, with whom nobody in that room except Murph actually has a relationship or emotional connection. Whichever way you come at it, it makes perfect sense for me.
Remember the joke line in
Tenet about needing to stop thinking linearly? Well, in a similar vein, you need to stop thinking in only our dimensions
In all seriousness, your confusion here is predicated on your not being one of the five-dimensional humans who evolved to the point where they could harness space and time the way that they do and create that tesseract. "They" opened that tesseract inside the wormhole for Coop. (Don't ask "how?" I have no answers for you there. This is just the mechanism that makes the plot make sense according to its logic, not necessarily to scientific fact outside the science fiction context.) Think of it like dropping a cell phone down to someone trapped in a cave. Silly example, I know, but it's similar to what's happening in that scene. As for what Coop does inside that memory box, time wasn't an issue. He wasn't playing Beat the Clock in that room. This is relativity again. Beyond the issue of time, which is a non-issue in the tesseract, the
real issue is
whether he can figure it out.* It could've taken him 10 seconds or 10 millennia "normal" time, but with the way that space and time have been harnessed, it'd be perceived and experienced instantaneously in Murph's room. (Time and space are relative, not absolute. That's key.) For another Nolan example, it's like the bomb going off in
Oppenheimer but it taking a while for the sound to reach them. Light travels faster than sound, so they see the explosion before they hear it. It'd be wrong to say that they "happened" at different times. This is just a function of our perceptual apparatus. The light and the sound are from the same source and occurred at the same instant, the sound is just slower getting to us than the light. Same thing with Murph and Coop. It's not that Coop only has for as long as little Murph is looking at the books, or big Murph is holding the watch. Time (the year and age she is and the year and age he is) and space (where she is in her room and where Coop is behind the bookshelf in the tesseract) are relative. It doesn't matter how long it takes Coop to figure it out, it doesn't even matter how long it takes him to transmit the morse code message to the watch, what matters is that he figures it out and figures out a way to get the information to Murph. It's like "posterity" in
Tenet: So long as it happens, it'll always happen. There's also more shit on gravity having worked on an object thereby transforming the object itself, so whether Murph has the watch in her room or in the NASA center, that second hand will be transmitting that morse code message forever because that's what Coop made happen "in the past" so that it'd happen "in the future."
As for Murph, it's not "randomly" that she figures it out. First, she'd always been the smartest one in the room. She knew Professor Brand was fucking up with his silly equation way before he admitted he was lying the whole time. Ever since she was a kid, she'd been "making fools of her teachers" as John Lithgow says. If anyone was going to figure it out, it was going to be her. "They" knew that, which is why "they" chose her, and "they" used her love for her father as the bridge between time and space. Now, it would've been much easier if Coop could've just yelled loud enough for Murph to hear him from inside the tesseract, or if he could've punched a hole in the tesseract wall and walked into her room in the past when she was a smart adult, but that's not how things work in the universe that Nolan established. Gravity is the most powerful force in the world, so he's able to knock books off the bookshelf and have them fall to the ground, he's able to use sand and wind to make coordinate patterns on the floor, and he's able to affect the watch. Second, as Coop tells TARS in the tesseract, she's going to come back for the watch because he gave it to her. Love is the bridge, it's that powerful force that transcends space and time, and it was powerful enough that even though Murph felt so betrayed by her dad and hated him so much for leaving, she was of course going to end up back in that house, in that room, and with that watch.
*A clever
emotional bit is that before Coop figures things out, he's all in his feelings. He doesn't see himself and Murph in the past, see moments that he lived replaying like bad TV reruns, and instantly become rational science man. He sees the past and
wants to go back, he wants to change the past
for himself, he tries to get himself to stay (hence him spelling out "STAY" with the books). When Coop leaves and Murph is crying and giving him the cold shoulder, he says "Don't make me leave like this," but when Coop is in the tesseract watching this moment in time he's shouting at Murph "Don't let me leave." I'm welling up just typing this. That's some powerful, deeply
human shit right there. He's a man out of time, a man with no place, truly floating in nothingness. He's made his choices, he's lost that time and he can never get it back, but he can make the ultimate sacrifice and accept that if it means that he can change time and space for his children, if he can do what he set out to do and save them, even if it means he loses everything. So he cries, he yells, he bangs on the wall, he gets out all of his emotions, and then all that's left is the mission, which he successfully completes. Heavy shit, man, and so devastatingly beautiful.