This isn't a satisfactory explanation either, I'm afraid.To me, it always seemed fairly obvious that a requirement of the magic is that she cast it from three different times--a past, a present, and a future--in order to set the gears of Time Compression in motion. Time Compression did take time, and maybe it would have been faster if she'd gone to the beginning and then to the end, but.
If you use the string example, it does hold up--you take three points on the string, and yank them together, /hard/, pulling the other ends of the string together--the momentum, and her continued use of the magic once it starts, keeps Time Compression going. Until, that is, SeeD kills the heck out of her and it starts to shift back to the way it was.
Firstly, your statement that she needs to cast it in the past, the present, and the future all at once is meaningless, because it singles out the time of the game (ie. Squalls era) as being the present, Ultimecias time the future, and pre-Squall time as the past. Such a division of time is quite arbitrary though, and there is no meaningful reason as to why Ultimecia would need to set Squalls time as 'present'. Much more reasonable is then to say that Ultimecia's present is the present in question, but since she never travels to the future, the idea that she casts it in past present and future doesn't hold.
Secondly, it is specifically made clear that Ultimecia has to be beyond a certain point in the past (as seen from her perspective) to cast it, but there is absolutely no logical reason as to why she needed to be at one exact location for TC to work. If she simply needed to cast it at two (or three) different times, there should logically speaking be no difference between any two (or three) arbitrary points, rendering this idea meaningless.
Thirdly, absolutely nothing indicates that Ultimecia casts the spell at more than one point in time! It is an assumption on your side which has been placed into the game with no basis whatsoever. Clearly this greatly lessens the plausibility of your idea.
Finally, there is a huge flaw in your concept of yanking time together like a string. If TC initiates an actually physical deformation of time itself, once time reverts back to normal after beating Ultimecia, there would be nothing indicating it ever happening! Think about it; it makes no sense that time itself is yanked together, so as to then revert back to normal afterwards, because it treats time itself as something which moves through time! (if it undergoes change, it must move through time). That clearly makes no sense whatsoever. Furthermore, if TC was an impermanent feature of the line of time, all the events which occur as a result of TC would only exist as long as TC does, which completely contradicts the games concept of fate and static time.
In other words, if anything is to make any sense at all, TC must be seen as an actual event on the line of time itself, and the gradual compression of time would be reflected within this single TC event, until everyone/everything emerges on the other side when TC ends.
For a detailed explanation on this concept, as well as more on time compression and time in FF8 in general, please check out the "Time/Ultimecia Plot FAQ" found here:
http://www.gamefaqs.com/console/psx/game/197343.html
It should hopefully explain things in a satisfactory manner. If not, please ask me here.
EDIT: I should say that the image of the string can still be useful in terms of understanding what TC actually accomplishes, but it simply cannot be used as more than a metaphore. Personally, by the way, I like to use the metaphor of considering time as a film, and TC as placing all the slides of the film on top of eachother to create one single image.