Who says they didn't? 

Y'see the problem is, before you had sheetmetal ductwork, they called it something else...not duct tape.

The ancient romans used it to attach limbs and noses to their marble statues.  But since it was made from parchment and gum arabic, rolled around wooden cylanders, it rotted away leaving ancient roman statues, limbless and deformed.

You mentioned W.A. Mozart, who was an avid user of quickie repair tape or Luftschacht-Klebeband.  The text from Ode to Joy was originally referring to feelings of elation upon fixing a leaky tap which was causing some irritation during a late night boozing and tabulature session.  His onstage Harpsichord was an older model that he referred to as "Trigger."  The finish was worn off and many of his lesser known contemporaries had written on it, part of the lid was actually held on with a classical version of duct tape. He commonly wrapped his upper thighs in it as well to prevent chaffing while seated at the piano for hours on end.

Hans Klemper, little known Viola player who was ALSO Debussy's Road Manager stated that Bande De Conduit and the black version commonly referred to as GAFF was used by Debussy's touring band during it's wild summer outdoor tours.  The heat and rhineland humidity would often warp the bindings of the principal violinist's Italian Stradivarius...or Strad, as the players commonly call them.  So rather than send them back to the Artisans for costly repairs, they would simply tape the joints down to keep them from buzzing.  This also helped on days when they played matinee shows and the Cellists smashed their instruments at the end of an encore of Blue Danube.

In the years before bumper stickers, Indie-Composers used duct tape to advertize their Orchestras.  Rather than costly ad campaigns, they would take strips of duct tape and write the names of groups on them and stick them on Carriages, Violin Cases, Chamber Pots and wherever else you could find to place them.  This is how the Austrian Baroque Sensation Johann and the Scorchers ended up on teenagers book bags and notebooks all across europe.

In the 20th Century, Aaron Copeland's concert Pianist Walt Klarner used Duct Tape to keep his charts on the stand during outdoor concerts.  And it is rumored that Copeland had other recreational uses for Duct Tape as well..but that's another episode.

So...it's not a new thing. 

PRH