I don't think Plato is popular simply because his work survived in more than fragments; rather, the reason so much of his work survived is that he was popular, so people made lots of copies of his books.
The other type of explanation would point at the social environment. Ideas fall on fertile or infertile ground because of what is potentially popular in a given culture / society.
In this domain, I think Plato played exceedingly well to an audience which sought to distinguish itself as an elite from the general democratic mob, but which considered both materialist showings-off and religious roles as not viable. Plato`s political theory, his anti-empiricism, the whole Socratic aversion against (the highly pragmatic / useful) sophists, those were things some people were bound to like.
Plato certainly had an audience from the start. Keep in mind that he was part of a tradition that defended elitist rule; that gave him popularity among a certain contingent by definition. But also note that his whole theory of forms was hardly what attracted his main audience initially. Platonism was popular, then declined in popularity (and stoicism became a big hit), and then became popular again with the rise of Neoplatonism and its more mystical elements-- a rise that largely coincided, not coincidentally, with the rise of Christianity. Gnosticism and such things happily borrowed from Plato. But the thing to note is that Plato was essentially 'rediscovered' in that second period. In between, he was hardly forgotten, but not wildly popular, either.
Compare this to Aristotle, who was somewhat 'neglected' much sooner (his school hardly lasted, while Plato's thrived for quite some time), and was 'rediscovered' much later as well. (Scholastics, borrowing from islamic sources.) In modern times, Aristotle is seen as just as much of a giant as Plato, but in the first ten centuries AD, I doubt many people would see it that way. This sort of implies that in order to be remembered, one doesn't have to be universally and lastingly popular.
I'm trying to say: we shouldn't overestimate Plato's initial popularity. He only became the big hit we know him to be when he was rediscovered later on, and Neoplatonism emerged. Plato's absence would have certain major implications early on, but once we get to the point where Neoplatonism emerged in OTL, the changes will get truly vast. Plato's influence was huge. The way Christianity puts such extra focus on the fall from paradise and orginal sin? Reflect Plato's idea of an ideal world, and this world only being a pale shadow. The influences on Gnosticism are even more evident. Hermeticism. Manichaeism. Mandeism. All the way to the Cathars. Later semi-mystical groups, like the Rosicrucians.
No, I'm not going all conspiracy theory here. I'm not saying there's a direct line between all of these, some kind of secret 'Platonic underground' of mystics (that conspiracy theory exists, by the way). But these movements did influence each other, and they were all influenced, at their base, by Plato. His absence fundamentally changes the way people are going to look at mysticism. There were earlier mystics, of course, but the whole theory of forms was a
very powerful influence.
So I dont think...
Now, without Plato, it`s likely that we don`t end up with exactly the same philosophy, but it`s also unlikely that we end up with totally unrecognisable systems of thought.
...that this is true. I think that especially later on, the absence of Plato is going to have enormous consequences. Besides the mystical elements, there are political consequences. As I mentioned, Plato was part of a tradition that defended elitist rule. That's the same tradition that idolised Sparta, and the idea that the state should be the supreme good. Again, there is no direct link, but Plato's political ideas were brought up again and again in later ages, and rarely to good ends. Because politically speaking, Plato was a scary totalitarian.
Inasfar as that is concerned, I'd say we'd be better off if his works had been forgotten. But then, I do maintain that without Plato, we'd have no Aristotle (at least not as we know him), so that's pretty terrible, from my Aristotle-loving perspective.
Another thing to bear in mind -- if Aristotle is sufficiently butterflied, that will affect the education of the boy who OTL became Alexander the Great; so potentially some pretty massive short term butterflies.
At the very least, the absence of Plato would alter Aristotle's life to such an extent that his tenure as Alexander's tutor is almost certain to be butterflied.
I think I prefer to suffer the existence of Plato's more horrid ideas, rather than face a world where neither Aristotle nor Alexander exist as I know and love them.