"the traditional songs we know must have been those which were popular enough to stand the test of time"

We should not confuse "traditional" with "popular".   Many of the traditional songs that we now know and have learned were not necessarily popular songs of their time. Collectors "discovered" these songs by searching the backroads of our country and found songs that may have been only known to a single family. After they became published and sung by others, then the popularity spread.   Even the more well known songs have so many different variations oweing to the oral process.