I would love to just write this off as "the writers wanted a good goodbye scene" since quantum teleportation in the temporal dimension(time travel) is really not my forte when it comes to quantum mechanics. However, by the nature of this site, I will try to answer your question. This is going to be long, bear with me. I have included several questions you might have after reading it, if I missed out anything, feel free to clarify.
Before I re-answer, let me tell you a short story. After a few hours of discussion with my colleagues, we could not arrive at any conclusion despite bringing in several plausible theories. Then, out of the blue, our professor (who is like nearly 60 but still an avid anime watcher) just walks over and proceeded to blow our minds with one sentence "It's a cross-time temporal loop." Then we proceeded to face palm ourselves for our stupidity in over-complicating the situation when it was really so simple. Side note: my theories were not wrong, but the factor I inputted into them were wrong.
Firstly, you need to understand how their time machine works, cause we underestimated its capabilities before. Their machine is not a linear time travelling device, it is a trans-spatial time machine. When they travel through time, they possibly travel to another universe as well. Believe it or not, it is actually theoretically easier to make this type than a linear time machine cause time flows relatively, thus theoretically, in a 'space pocket' between universes, time would pass differently for separate universes.
But I digress, thus, Suzuha did not actually travel from the future, but instead from a potential future of the Steins Gate World Line. Now, Okarin never actually needed to go to the Steins Gate World Line, he left it as he saw Kurisu dead. But in reality, he was there in the beginning, he then proceeded to send himself all over the time lines with his microwave oven when he sent that mail to Daru.
What is the loop? The time loop created works like this:
- Okabe sees Kurisu dead, send mail to Daru, enters other timelines
- Okabe spends 3 weeks in alternate timelines, gets back to Steins Gate
- Time machine shows up with Suzuha saying they have to stop WW3
- Okabe goes back to past, fakes Kurisu's death, burns thesis, stops WW3
- Past Okabe sees Kurisu dead, goes back to step 1 and repeat
Remember the Attractor Field-World Line theory employed by the show, now imagine 2 lines intertwined but splits like a split hair at a certain point. To both these splits, their past is the same, but some event split this time line. This event was Okarin gaining the knowledge that he saved Kurisu.
Despite the aim of going back to the past was stopping WW3, we can never really say that WW3 was fought over Makise Kurisu's Time Travel thesis completely cause we have too little details. A more likely reason would be Okarin built the time machine based on her thesis using the knowledge he gained from the other World Lines and this information was obtained by world powers and they started the war to obtain this thesis to built their own time machine as Okarin was already dead in 2025.
The motives for WW3 was effectively instigated by Okarin. Okarin built the time machine to save Kurisu and thus as long as he does not know he saved her, the time machine would still be built, WW3 would still occur. That is essentially why Suzuha cannot disappear until the Okarin that knows he has saved Kurisu appears in the present again, or Okarin would go on to create the time machine to save Kurisu and start WW3.
But, to save Kurisu, the time machine is necessary, if Okarin did not have the motive to make it, where did it come from?
This is where the cross-time part comes in. Remember that I said Suzuha came from a possible future of the Steins Gate timeline, a split in the World Lines. This is where quantum causality comes in, as long as there is a chance, WW3 can still occur despite the thesis being burnt as Okarin can still make the time machine. The core reason for this as I mentioned is because he thinks Kurisu is dead. Remember that the first time he went back to the past, he stabbed Kurisu himself, no World Line change occurred, that means that they were still connected to a timeline that leads to WW3. However, when he saved Kurisu and burnt the thesis, still nothing happened.
That is because at this point, the two timelines still share a common past, and Okarin still wants to save Kurisu as the him that saved Kurisu is still in the past, future cannot affect them and they cannot affect the future yet as they are in a state of temporal uncertainty by being in the past. Same as the first answer but this time the factor is not the burning of the thesis but Okarin returning to the present with the knowledge that he saved Kurisu and does not want to create the time machine anymore, thus completely erasing any minuscule possibility for WW3 to occur. This leads to Suzuha disappearing as the two futures no longer have a common present. This returning of Okarin to the present is the diverging point of this two timelines. Thus, Suzuha can no longer stay in Steins gate as this is not part of their shared past.
Then what was the point of faking Kurisu's death?
Simple, it was to keep the loop going, if he broke the loop, the him that is trying to save Kurisu would definitely disappear for he never spent even a single moment with Kurisu. No matter how far quantum causality goes, it most likely does not extend into separate timelines for his knowledge is not from the universe he is in. Thus, the loop has to occur, he has to keep the past somewhat constant until his return to the present with the knowledge he saved Kurisu, only then will the timelines diverge properly.
What would have happened if they stayed in the past?
This depends on which travel they stayed in the past.
In the case scenario where they failed to save Kurisu, Okarin would disappear at the divergence point. Since this Okarin never returned to the divergence point with the knowledge that he saved Kurisu, it would go the WW3 route, the past Okarin thinks Kurisu dies and went on to create the time machine, starting WW3. This allows Suzuha to exist. Since Okarin traveled to the past from the Steins Gate timeline, he disappears since that is no longer his past now.
In the case scenario where they managed to save Kurisu and stayed in the past, Suzuha would disappear at the divergence point. Despite Okarin never returning to the divergence point with the knowledge that he saved Kurisu, they would still arrive at the divergence point where they will witness themselves going back to the past and thus arrive at the same case where even without a time machine returning them to the present except that the moment they witness Okarin going back, Suzuha disappears.