Said another way - light follows the shortest path between two points. In flat spacetime, that is a line. In curved spacetime, it may be some other path - like a path traced by an ant walking on a sphere. These paths are called geodesics:
https://en.wikipedia.org/wiki/Geodesics_in_general_relativit...
Gravity can stretch spacetime to make the distance light has to travel longer than in an unaltered "flat" universe. But can it also shrink, causing a photon to arrive at a destination in a shorter time? And I don't mean a wormhole, which requires tearing in a topological sense.
Correct. However, the crucial part that is missing from your thought process is that: 1) light can only travel in a "straight" line, and 2) "straight" lines are defined by the spacetime metric.
As a lower-dimensional analogy, think of what a "straight" line means on the surface of a sphere[1]. The shortest path between two points is, by definition, a "straight" line, but if the space itself that the path is embedded in is curved, then when viewed from a higher dimension (i.e. on the surface of the sphere it's straight for you, but if you were in space) it would appear curved.
The analogy fails when you attempt to think about it in higher dimensions, since humans have a very difficult time perceiving anything greater than the standard 3+1 spatial & time dimensions. But the math still holds.
Light will always travel in a straight line, it just so happens that gravity redefines what "straight" actually means[2].
1. http://mathworld.wolfram.com/GreatCircle.html 2. Explanation simplified for clarity.