Why lasers for single mode? It’s not about the brightness, for you can get some seriously powerful LEDs. And it’s not about the size of the core, we have lenses for that. So Why is laser light used in single mode step index optical fiber?
It doesn't matter whether the fibre is step index or graded index if we want to discuss broadly about sources required in fibre transmission.
You will need a laser or light source with enough amplified emission to produce coherent light. Because Fiber isn’t perfect, it loses signal with distance, plus more for connectors, bends, etc. If you want to send your signal a long distance, you’re going to want a pretty bright light.
If you use LED as light source, your light will get attenuated at smaller distances in fibre. A source which has good directionality and a proper choice would be LASERs.
It's known that signal loss is a common problem face optical fibers, which is caused by many reasons, like distance, connectors, bends, etc. As for multi-mode fibers used for not long distance transmission, LED light sources can be freed. But for single-mode fibers, which are used for significantly long distance transmission, a laser is required for bright light.