Meet Vladimir Shukhov, a Russian architect who first developed hyperboloid structures. He was born in 1853, died in 1939, and created over 200 hyperboloid structures in the intervening years. He was the reason hyperboloids gained the popularity that they did. His first design, the first hyperbolic structure ever, was the Shukhov Tower in Polibino, pictured here:
Another tower also bears the name of Shukhov, and it achieved great fame, too. Shukhov also built the Adziogol Lighthouse. In total, Shukhov designed and built 200 hyperboloid structures. He died in 1939, which could one reason for the decline in the popularity of hyperboloids. It isn't common for the death of one architect to essentially stop an entire movement (although Gaudí also worked with hyperboloids), but that may have been the case here. Shukhov was the reason hyperboloids were popular in the first place; without him, their popularity died down. Also, part of the reason that Shukhov was interested in hyperboloid structures was that hyperbolic geometry was being developed during the time when he first entered architecture. A Google n-gram for "hyperboloid" shows that the term's usage during the late 19th century was its peak.
As a final note, many hyperboloid structures were built after Shukhov's death, so their decline wasn't that steep.