During the Nine Years War (1688-1697) England was allied to many countries fighting France. After a few years of peace the War of the Spanish Succession from 1701 to 1714 England and its many allies fought France and the French allies.
For 26 years from 1988 to 1714, except for four years from 1697 to 1701, English privateers could get licenses to capture French merchant vessels. Thus they profited greatly by looting French civilian shipping. There were also hordes of French privateers looting English civilian ships.
And then peace broke out in 1714. The probability that goods shipped by sea would reach their destinations skyrocketed. Insurance costs plummeted. Economies around the world grew as imported goods became cheaper. A golden age of shipping and trade began.
And the former privateers were horrified by those curses of peace, and their lack of opportunities to steal legally. So they decided to steal illegally as pirates, and to attack ships from all countries, including their own, to seek revenge on the ungrateful governments who had cut off their source of wealth with no better excuse than peace was for the greater good of the whole world.
During the war, English ships risked attack by French privateers. After peace was made, thousands of former French and English privateers turned to piracy, and English ships now risked being attacked by former English privateers.
So I don't think that pirates would have been considered a big bogyman by the English public in 1706, or until peace was made in 1714 and the so called "Golden Age of Piracy" began.
The idea that pirates would have been a major bogyman to the English public in 1706 is one of no doubt many things that puts Black Sails in an alternate universe where history was different, just like almost all historical movies and television shows.