"So this is another ballad that it is highly likely to have had its origins in the broadside press but passed into oral tradition"

One thing that argues against a broadside origin here is simply the chronology. Seems a suspicious gap of time between the events and their commemoration in song. We know that songs were a common method of passing news, and songs were typically composed or adapted (either by folk or by broadside writers) soon after the events they referred to. An obvious exception would be when broadside printers tapped existing songs; in the absence of a more contemporaneous broadside, which origin seems the more likely?

We also know, from this and other songs, that the names of ships and captains are particularly given to mutate over time, and moreso if the captain/ship are more representative than memorable. Ward was memorable; the men and ships he defeated were not. Ward was already over 50 and a veteran privateer when the turned corsair in 1603; his career continued for another 19 years, and he remained unapprehended even to his death. He must have defeated or chased off quite a number of English ships during that time. One of the ships he captured (c.1606) was named "Reniera e Soderina", and he used it as his own ship for a time; this could be the genesis of the name "Rainbow", even if factually it doesn't quite fit the encounter. The song's origin was probably close to 1607, when King James refused to grant Ward's petition for a royal pardon--the inspiration for the purported boast that he'd remain "king on the sea"?

Most of the early ballads, whatever their source, tended to be long and clumsy, just as most songs that now survive in oral tradition tend to be appreciably shorter and more polished. Fashion may have more to do with this than whether the songs began as hack broadsides.

I don't swallow the "Travellers were illiterate" line of reasoning, because they didn't live in a vacuum. Especially in recent times (but before their songs would have been collected), they were surely also learning songs from people who could read and who did learn songs from broadsides--and modifying their traditional songs accordingly. After all, the Travellers are known for how they adapt into each culture, and serve as conduits for intercultural exchanges.