It still didn't kill the gaming computer at all. After all, around the same time came the Amiga and Atari ST, and PC got more and more turned to gaming by getting EGA and then VGA graphics cards. Even Apple tried to woo gamers at the time. Germany and France for instance continued gaming on those machines and others (Amstrad/Schneider CPC, anyone?), and the tradition of computer gaming has survived to this day, especially in mainland Europe where PC gaming is still considered very big.
What it did, was bring in a new type of competition. But the NES (or consoles in general) had no chance at all to kill the PC gaming market of the time since they couldn't compete with the prices that were usual on PC outside of big boxes, especially cassette games were dirt cheap (10$ would have been considered expensive for those back then). Also, the PC had genres the consoles didn't have, or not in that form or size, like managerial games and adventures, both which got bigger and bigger in the late 80's and early 90's.
On topic: Where's Tetris???
The performances of the gaming market say otherwise as both NES and SMS chipped away at the computer market to the point that the SNES and Genesis outright ended that era until PCs took over, cassette games were dirt cheap but non of those games were anything like SMB, Sonic etc... which consoles were being driven by it was a complete shift in the market that ended with the likes of the old guard going under as they couldn't compete with the new consoles. To give you an example the Amstrad CPC sold 3m compared to the 6m of SMS and 8m of the NES highlighting my point as consoles at the time of the release of these devices were dead yet the home computers were beaten out in the end.