Taken together, this is an unlikely story told by a Middle Platonist around 100 AC: how Plato influenced the development of geometry in the first decades of the 4th century BC.
Elegance is a matter of taste, which is not the issue here.
In the 4th century BC we see an attempt to simplify geometric construction, that leads to results compiled around 300 BC by Euclid, and afterwards becomes the paradigm for the scholars of the following thousand years.
At that time many ways to double the cube with other means - a good summary is here - were known, though around 350 BC the dialogue Sisyphus claims the Delian problem to be unsolved.
This at least shows that its unknown author - likely from the Platonic academy, where the legend of Apollo's oracle to the Delians was told (see Kouremenos above) - does not recognize the solutions of Archytas, Eudoxus and Menaechmus. And this again hints at a motivation of Plutarch's texts above.
Both together strongly hint at the origin of the Delian problem in the second third of the 4th century in the ambit of the Platonic academy, and its exclusion of the methods of Archytas, Eudoxus and Menaechmus to preserve the purity of geometry.
OneShot, I think you are incorrect on a large number of points. However, I also think our discussion (albeit interesting to me) is getting seriously off-topic, so I won't debate on the history of constructive geometry any further here. Feel free to pm me if you wish, though.
That said, back to the OP!
I have to agree with Tellus, about the fact that even without building "magic computers", if one manages to "embody" mathematical problems into objects, it's easy to use the low level Intellego guidelines "learn one fact about ..." to solve said problems - sometimes quite a bit better than modern computers can accomplish. And it need not be geometry.
Here's an example. A faerie princess gave you an enchanted flute, a glamoured bag, and a quest.
The flute will play itself, and while playing it will keep the greedy dragon that sleeps under the mountain from waking.
The bag can hold up to half the dragon's hoard in weight, though no matter how stuffed, it always looks and feels a small, half-empty bag.
The quest is to bring back to the princess as much value of the dragon's hoard as the bag can hold before the sun next sets and rises again.
A little bird has told you that the princess assigns a very specific value to each object in the hoard, which is exactly equal to the amount of copper that can be smelted from it. However, the object must be retrieved whole, or the princess will call you a cheat!
Even if you know the exact weight and amount of copper of each object in the hoard, this is an extremely hard math problem! It's called the knapsack problem. Even the most powerful supercomputers of today cannot in general solve it in 24 hours, if we are looking at a several thousand objects. But an InTe spell cast at T:Group on the hoard can determine which is the set of items with the most copper in it, that still weighs at most half as much as the entire hoard.