Examples where insight from geometry was useful for solving something completely non-geometric
A few more examples:

Sleator, Thurston and Tarjan used a geometric representation of trees as partitions of polygons, and hyperbolic geometry, to prove lower bounds for binary tree rotation. (Also, I believe the history of a dynamic binary search tree can be represented as a tetrahedralization.)

The reduction of least common ancestor to range minimum queries, due to Berkman and Vishkin, relates a data structures problem on trees to an arguably-geometric problem. (and thanks for the article David)

The reduction of a scheduling problem to max weight independent set of axis-parallel rectangles [1] or the reduction of a different scheduling problem to geometric set cover [2] might qualify.

The reduction of the largest common subsequence problem to finding layers of maxima is well-known (meaning, I'm too lazy to look up who actually thought of it).

[1] (Liane Lewin-Eytan, Joseph Seffi Naor and Ariel Orda)

[2] Nikhil Bansal, Kirk Pruhs. The Geometry of Scheduling, FOCS 2010.

[later edit] A couple more cases where a "geometric" view seemed surprising (though the "submission to SoCG" or "makes something to visualize" standards are probably not met):

algebraic topology applied to lower bounds for distributed computing

incorporating computability into Hausdorff dimension

defining a notion of distance for groups, then volume, then growth of volume as a function of distance, then using "polynomial growth"