The short answer: yes, pirates constituted a "legitimate threat" to major sea powers; yes, their threat was overstated and sensationalized by various media of the day.

The most difficult part of your question is determining what "influence and power" means. Were there a significant number of active pirates during the golden age? The best research suggests that there were approximately 1,000-2,000 active at in the early 18th century. The British Navy alone outnumbered them, employing anywhere between 6,000-12,000 sailors during the period. Add in other maritime powers operating in the Atlantic and you'd have a much larger disparity. Did pirates cause a lot of damage? Yes, undoubtedly piracy caused significant economic harm to merchants and shipowners, although as this thread describes it is difficult to determine how much damage was actually done. Did piracy breed more piracy? Yes, there is considerable evidence to suggest that most pirates became pirates after their merchant ships were captured.

The best evidence for the "power and influence" of pirates during the golden age would be the state's reaction to them, and at least in the British context, the reaction was significant. Piracy was heavily criminalized, pirates were routinely tried and publicly hanged, and publishers (typically religious) would routinely distribute anti-pirate propaganda. This was especially true in the 1720s, when close to 1 in 13 known pirates were hanged by the British. (Rediker, Between the Devil and the Deep Blue Sea, (Cambridge UP, 1987) 283n97). The extraordinary state effort to rid the sea of pirates, then, is evidence of their influence.

Now, as /u/davidAOP notes in linked thread, there isn't much evidence to suggest that the damage done by piracy was greater than the damage done by various interstate conflicts throughout the long golden age of piracy. So in a sense the "legitimacy" of the threat is not that great. However, it's important to remember that the threat of piracy is different. Because pirates are by definition non-state actors, the threat they pose to the state is at once universal (i.e., they can attack the ships of any state, although in practice they often did not) and enduring. As Daniel Heller-Raozen puts it: "By virtue of their enmity with respect to a general collectivity they fail to constitute an association with which there might be peace as well as war." (Enemy of All (Zone Books, 2009) 11). In other words, pirates pose an existential threat to the state and must be dealt with differently than a competing state actors (e.g., the relations of war). This observation pushes the boundaries of traditional historiography into the world of political theory, but I think the exceptional nature of piracy in comparison to other states of amity is important.

My expertise is in 19th century Atlantic piracy, so if you have questions about that period I can provide a much more in-depth answer.