The Broadway musical was the main conduit for "success" before film and TV. The writers who had success there had a running start, which created a business apparatus to lock it in. That apparatus continues to this day
Outside of that realm, though, you have Tin Pan Alley and the song pluggers. All you have to do is look at the collective early recordings of, say, Billie Holiday to see that most of those songs were drek and failed accordingly.
And really, of those writers who had hits outside of the world of musicals, how many had LOTS of hits.
Success feeds on itself, but until the business comes along to entrench it, that feeding is usually going to be a snack.
I think the better question is not who gets to write a standard, but who gets to own one after it is written. That is who will ensure that it stays one.
Staying one...I just looked at a Wiki list of Walter Donaldson songs, and most are forgotten today and the ones that aren't are the sort of songs that you are going to have to have done your homework to know about.