Digital piracy took off in part because copyright restrictions were so unreasonable and onerous to begin with. Copyright was last extended to its current absurd length in 1998, and its provisions were retroactive, which is arguably unconstitutional.
I agree that current copyright terms are too long, but for your causality argument to hold water, the bulk of piracy would have to occur for works older than 17 years (the initial copyright term), and I very much doubt that this is the case.
I think it took off because we finally saw behind the curtain.
By and large, people could ignore copyright for its first few centuries-- it simply wasn't in the face of most people.
In 1799, or 1899 you couldn't reasonably print a commercial-quality newspaper, novel, or sound recording at home. If there was anything resembling commercial-scale copyright infringement, it was a business-versus-business play, and usually the 'knock-off' product was an obvious inferior substitute, or a clear attempt to defraud buyers. You weren't just paying for the "legitimacy" of official works, you were paying for a product higher quality than you can generate yourself.
By 1999, a normal upper-middle-class consumer could buy some sub-$1000 printers and produce his own copies of documents and photographs that looked as good as anything you got from a publisher. He could get a spindle of CD-Rs in a neighbourhood shop and make audio CDs that played as well as commercial ones.
At that point, the obvious question is "this costs me 75 cents to make at home, what am I paying $15 for?" The answers-- "it's legitimate" or "the artist gets paid, blah, blah", did not prove satisfactory enough. The "artist gets paid" one especially rang hollow when we saw plenty of artists sell millions of $15 records and earn enough in royalties to get a Big Mac but not fries.
Only the fear of prosecution was remotely viable to force people to ignore the question and go back to buying "official" media.
The sad thing is that the 'legitimate' options we have now are arguably even worse - streaming services pay laughably tiny royalties, there's no transparency or consistency, ad supported platforms are markedly worse than broadcast radio/tv were.