Well, I didn't say flashbacks were the only way to do it. Merely A way.
Flashbacks may be more effective for some characters than others. The Flash, for example, has a much simpler origin than, say, Wonder Woman. All you need to do is show Barry working in a police forensics lab, depict an accident involving electricity and chemicals, watch him begin to discover his abilities, and then flash forward (ahem) to Barry in the costume.
Anyone who got their powers through a singular event can be taken care of this way. That lets the writers reserve more story time for characters whose backstory is actually part of their identity and needs to be told — Wonder Woman, Hawkman, Batman (if it's still deemed necessary)....