Good for you /u/SpikesHigh for questioning where the pirate stereotypes come from. Though, mind if I dissect the ones you listed?

• Captain Kidd and buried treasure - yes, that did play a big role, though Robert Lewis Stevenson and the story of Treasure Island appears to be the thing that nailed home the concept for many. If anyone is interested in why finding buried treasure got so popular in the first place, the 19th century is full of stories of lost valuables being buried and people seeking them. It was a cool thing to add to folk lore. There is an article on the subject entitled "The Early Republic's Supernatural Economy: Treasure Seeking in the American Northeast,1780-1830" by Alan Taylor from the American Quarterly if you're interested in the treasure hunting obsession in America.
• Blackbeard, long beards and tricorns - yes on the beards, and no on the tricorns. Tricorns, or known back then as cocked hats, were not necessarily a "thing" for pirates or sailors (the two overlap heavily) to wear. Wool caps of various kinds (in particular caps like the Monmouth cap) and felt hats with mostly unshaped brims (often mis-shaped in some non-uniform way) that were also called round hands (since the crown of the hat was completely round, though sometimes they had flat tops) were the dominant hats. Cocked hats didn't become a fashion for sailors until by the 1730s, after the Golden Age of Piracy. That doesn't mean a pirate didn't pick up a cocked hat now and again, but it wasn't prevalent. As for who popularized it - it's hard to nail it to any particular person or piece. A general trend is that, over time, artists either copied fashions of sailors contemporary to them - or by the end of the 19th century picked up sailor fashions from the late 18th to early 19th century and adapted them to pirates (since access to sources describing the later sailors are more regularly available and many assume that sailors in 1715 looked just like those in 1799). Then, add onto that Howard Pyle's huge art influence that decided to mix in 19th-century Spanish colonial dress and gypsie-ish dress into the pirate look to help differentiate his pirates from other men of the era.
• "Calico" Jack Rackham and the skull and sword [flag] - yes, that is attributed to him, but historically Jack Rackham is never documented flying that flag. See this discussion for more about how messed up pirates flags became because of unattributed illustrations of the early 20th century.

Now for the parrots - again, Treasure Island is responsible. In the realm of influencing our modern perceptions of pirates (that isn't also technically a period source, that award would go straight to Charles Johnson's General History of Pyrates, 2-volume worked published from 1724-1728), there is almost a tie between Stevenson's book, Howard Pyle's art in the late 19th and early 20th century, and Disney from 1950 on. In Treasure Island, Long John Silver has a parrot called Captain Flint that says "Pieces of Eight". Before that, pirates didn't have any particular stereotype for parrots - just like they didn't have any particular stereotype for wooden/missing legs, hooks for hands, or eye patches. Sailors of the Age of Sail were known to occasionally acquire animals in foreign ports through purchase or trade. Some may have kept them as pets for themselves, but at least with the case of parrots, they were also excellent profit to sell back in non-tropical ports (especially Europe).

On a related note (since I mentioned missing appendages), while it's probable that Treasure Island really helped push the image of pirates as being disabled in some way - as far as I can find, the stereotype probably has origins in the adaptation of the late 18th/early 19th-century stereotype images of Greenwich Pensioners, disables sailors taken care of by the Greenwich Mariner's Hospital. They are often pictured with patches, hooks, wooden pegs, or missing limbs aided by crutches. They often have remarkable resemblances to the pirate stereotype. Life at sea was dangerous and accidents happened. It wasn't limited or special to pirates, though pirates would have suffered too.

Hope that answers your question and clarifies the origins of pirate stereotypes more. Besides the first chapter of David Cordingly's Under the Black Flag: The Romance and the Reality of Life Among the Pirates doing a good job at explaining the origins of pirate stereotypes, the literary legacy and it's impact on piracy is also explained in scholarly fashion in Neil Rennie's book Treasure Neverland: Real and Imaginary Pirates. For the concept of Howard Pyle's influence on the pirate image, my first contact came from conversations with a man named David Rickman, who eventually put it to print in his work with Angus Konstam in the Osprey Publishing book Pirate: The Golden Age.

EDIT: A friend of mine reminded me that one potential ("but how are you ever going to prove it" kind of situation) inspiration for pirates with missing legs/wooden legs is from Charles Johnson's General History of Pyracy, in particular the Captain England chapter where there is a, "Fellow with a terrible Pair of Whiskers, and a wooden Leg, being stuck round with Pistols, like the Man in the Almanack with Darts..." He's noted for standing up for a merchant captain they captured and saying he was a good captain and shouldn't be harmed. It seems like a great inspiration for future one-legged and wooden legged pirates, but it's one reference in one chapter of this book. While it probably inspired some, unless a writer specifically said, "I was inspired by this piece in this book," it's hard to say if they were or not.