Pirate-themed birthdays, pirate halloween costumes, Captain Pugwash, Captain Jack Sparrow, this song, and a long list of other pirate-themed children's entertainment would likely be extremely jarring if a time traveller from the 1700s saw them. When, why and how did the image of pirates become so much softer?
The roughly 200-year evolution of pirates into their current iteration of kid-friendly adventurers and villains can largely be attributed to three books, one stage play and novel, one American illustrator, and two films:
Robinson Crusoe, 1719, Daniel Defoe
A General History of the Robberies and Murders of the most notorious Pyrates, 1724, Captain Charles Johnson
Treasure Island, 1883, Robert Louis Stevenson
Peter Pan; or, The Boy Who Would Never Grow Up, 1904 stage production & Peter and Wendy, 1911, J.M. Barrie
Howard Pyle, 1853-1911, Influential American Illustrator of seafaring adventurers
Treasure Island (film), 1950, RKO-Walt Disney British Productions
Peter Pan(film), 1953, Walt Disney Productions
Robinson Crusoe is as definite an origin of the transformation as any, especially since it's not directly about pirates. This work was so wildly popular the first volume had four editions in its first year of print. The desert island adventure trope was thrust into the zeitgeist and Crusoe spawned it's own genre of copycat works, "Robinsonade."
Following Crusoe by five years, we have A General History of the Pyrates by an enigmatic (probable pen name) Captain Johnson. Here we have another well-selling book about (likely exaggerated) adventures on the high seas featuring the Jolly Roger flag and many other household names and tropes that survive in popular fiction to this day. Pyrates was a major influence on the previously mentioned authors Stevenson and Barrie.
Treasure Island is our first major turning point, towards "funny characters for kids." This was a serialized work that first appeared in Young Folks (1871-1897) a magazine for children. Similar to Pyrates it was published there under another nautical pen name, Captain George North. Treasure Island, again, has many a pirate-ical trope that survives today such as the missing leg, animal (parrot) companion, X-marked treasure maps, and dessert islands. The short answer to your question is undoubtedly "1883" with the release of Treasure Island when it reached popularity in book format.
In the years following the release of Treasure Island we have a series of illustrations by Howard Pyle who is largely credited with popularizing the look/dress of the modern pirate. Many of his works are currently on display at the Delaware Art Museum: https://delart.org/collection/american-illustration/
The origins of pirate-talk and how pirates carried themselves can be traced back to the 1904 production of Peter Pan at the duke of York's Theatre in London (and the subsequent novel) which was later adapted into the more popularly performed pantomime version intended for families and children.
The final transformation of the pirate into what we see today in children's literature, television, and film can be largely attributed to the Disney productions of Treasure Island and Peter Pan. The former was a modest success at the box office, benefitting from a US rerelease in 1975. Peter Pan, however, was and remains hugely successful, with a lifetime domestic gross of nearly $90m.
Further reading:
Cordingly, David (1995) Under the Black Flag: the romance and reality of life among the pirates
Watt, Ian (1994). Robinson Crusoe as a Myth. Norton Critical Edition
"Collected Letters of Robert Louis Stevenson"
Green, Roger Lancelyn (1954). Fifty Years of Peter Pan