As with all eras, there's always a backlash against sounds and genres as they become overplayed and predictable, and people start cashing in on it rather than making it for the love of it. I don't think it's entirely fair to pin it on one instrument as being the cause of it, especially given how well some of the songs made with it have weathered the test of time.

Digital sounds in popular music went through that backlash in the 90s and early 2000s. Nothing could have prevented the backlash, and no one thing caused it. It happened to disco in the 80s and 90s when it was the butt of jokes, which then saw a solid resurgence in the 2010s with the pinnacle being Daft Punk making what was effectively a disco album.

DX7 was responsible for more hit synth sounds in the mid to late 80s than any other synth (probably, though there were some other contenders, and a few analog signal path, digitally controlled, synths that definitely had a big impact during the same era...the Prophet and the Roland JX-8P come to mind as instruments that would have often appeared alongside the DX7 on record). And, many of those songs are still loved and played today...and often continued to be played throughout the anti-digital era of the late 90s until the early 2000s when we started seeing EDM producers and others begin to reintegrate FM synthesis and other digital techniques into their sound (dubstep is all about FM synthesis, for example...those big angry swooshy wubby noises are often FM synths).

And rightly so: the DX7 presets were abominably tacky.

Only in the hands of a capable programmer can it go to places unreachable by contemporaneous machines. Eno is the most obvious name that comes to mind... do you know any others that plumbed the depths of FM?