Whether you could've done the same with a strop, depends on the strop, and also on why the edge wasn't shaving hair initially. If the poor shaving was just due to a wire edge that had rolled over, a strop could probably fix that. If, on the other hand, the lack of shaving performance was due to a 'dull' edge, or a slightly less-than-perfectly apexed edge, a strop might not fix that (probably won't, most of the time). A bare leather strop won't do much on an edge that isn't truly ready for stropping (meaning it's not fully apexed). And a strop with very fine compound, or a compound not ideal for the steel, might not help either.
When I 'touch up' my edges, if I find that the strop isn't quite enough, I'll take a very small step back in grit. For example, if I usually strop on bare leather, and need something a little more aggessive, I'll fall back to 1 micron diamond paste on leather (or hardwood). If that's not quite enough, I might step back to 2000 grit wet/dry sandpaper on top of my strop block. That usually is enough to get the edge popping again; then I'll go back up the grit chain, to diamond, then finishing on bare leather or on my jeans.
With time and experience, you'll begin to recognize, just in the way the edge is cutting, how much exactly is needed for a touch-up. It'll minimize the 'overkill' on sharpening a blade, so you don't remove more metal than is truly necessary. If you strop or touch up fairly often, and don't let the edge become truly dull, you'll rarely need to go back to the stones, or at least, never back to the coarse ones.
