Why, the Atari Falcon030 killed the Amiga of course. :-D (just kidding)
Well, as someone on the ST side of the fence in the era of the great machines, but a big Amiga fan too, the thing that killed off the Amiga is the same thing that killed off the ST -- bad management.
Both Atari and Commodore were WAY ahead of their competition when the STs and Amigas shipped. In 1985, colour graphical computers with multi-voice sound, for less than a base B&W Mac or green-screen IBM XT, were incredible things.
The stuff you could do on either was amazing. Each had its strengths -- the Amiga had incredible sound and colour palettes, the ST had OK colour and sound and a stable hi-res display that didn't flicker at 640 x 400.
The applications for each were ASTOUNDING compared to their contemporaries. By 1987, just two years after introduction, both systems were in their prime. They had apps that put the PC and Mac to shame.
But then what happened?
Nothing.
Atari and Commodore both sat on their laurels and released slightly souped-up machines that didn't address market needs. When I think back to the lost opportunities, I want to cry.
Commodore released the A500, which was a cost-reduced version of the 1000, but otherwise unchanged.
Atari released the Mega ST, which was a recased ST with a blitter and more RAM.
Meanwhile, really really cool apps on the ST moved to the PC, since Atari didn't deliver on its promises for faster hardware, math coprocessors, etc. The AutoCAD software of today grew out of the Cyber Studio on the ST -- if Atari had kept pushing development and come out with an 020 based box (fully possible in the late 1980s) with VGA-type colour palettes at 320 x 200, it would have kept its lead in this area.
Oh, and Commodore came out with the A2000 -- more expandable, but that's about it. And the price was sky-high!
Fast-forward to 1990 or so. Atari had the STE which did little but add extra colours to the palette and SIMM-expandable memory (whoopdee doo!). The A500 was still. . . the A500.
The TT030 and A3000 were both cool machines, but didn't really advance either company's lead and were poor sellers.
By the time the Falcon and A1200 shipped, both companies attention was elsewhere. Commodore's management decided that it could make a killing selling PC clones with no effort at all -- a decision that resulted in losses in the hundreds of millions of dollars when Amiga sales were white-hot.
Atari released the Falcon030, which technically outpointed every other system out there (including the 1200, sorry Amiga partisans ;-) ), and proceeded to SCREW the remaining dealers with no advertising, as it decided to dump computers and focus on the Jaguar.
So Atari abandoned the computer biz, and Commodore allowed the Amiga biz to float with no real direction, as it kept trying over and over to make it big in the PC clone biz.
Commodore ended up getting slaughtered by Dell and Gateway and those guys in the PC biz, while the Amiga languished. The 1200 and 4000 were both kludges rushed to market in an effort to get something, ANYTHING out -- they were NOT where Commodore's engineers wanted them to be.
AGA had been sitting in Commodore's labs for YEARS by the time it shipped in 1992. In 1990, it was earth-shattering and would have buried Atari and Apple both. By 1992, it was old hat -- both the Macintosh and Atari were ahead.
The 3000+ had an AT&T DSP that would have delivered a great degree of sound to the average Amiga user, had Commodore shipped it in the 1200 and the 4000. Instead, they shipped the then-approaching-obsolete Paula, while Atari shipped the Falcon with a DSP and Apple introduced CD-quality sound as a standard feature.
The remarkable thing is that the Amiga thrived through all this inattention. When Commodore went under, Amiga demand was at RECORD levels, despite the severe channel, advertising and product release mismanagement. However, Commodore was so indebted and starved for cash that they couldn't produce enough machines to meet demand and turn a profit and pay the bills. Had Commodore decided in 1990/1991 to focus on the Amiga alone and ride with it, and push forward with the move to RISC (with HP as a partner), the Amiga would have survived and thrived, and Commodore would still be in business today.
The other thing that's killed the Amiga are all the false starts post-Commodore. Commodore and the Amiga, like it or not, are part of each other. Without one, the other cannot survive.
I know there are new clones/machines coming out, but the ST scene saw those too, post-Atari, with C-LAB building Falcons and other companies providing super-advanced clones and new iterations of TOS, but it didn't save the ST scene because Atari's distribution, marketing and name were necessary for success. Ditto for the Amiga. Now that the Commodore R&D, distribution and marketing infrastructure is gone, Amiga pretty much is too. Which is sad because the ST and Amiga were so much better than the Macintosh (let alone Windows) and I cringe to think of how much better the Amiga and ST would have done things if development had continued from the early 1990s on.