I think that Gauss Theorem on constructible polygons fit this cathegory.
For more than 2000 years the actual construction only lead to 4 classes: $2^n; 2^n\cdot 3; 2^n \cdot 5; 2^n \cdot 15$.
Gauss' abstract aproach solved the problem. The interesting case $n=17$ becomes easy to understand, and easy to construct once one understands the abstract approach, but hard to attack otherwise. $n=257$ and esspecially $n=65537$ and the ones derived from these are the perfect examples of easy abstract proof vs. extremelly complicated calculations.