Back then the Clintons were positioning themselves as the moderate Democrats, attempting to build a bridge with Reagan Republicans since Reagan was still a widely beloved figure at the time. The general push towards neoliberalism in general meant that strategically speaking, the Clinton Dems understood the need to not make enemies of Reaganites.

Going after video games was a clever way to pander to social conservatives without attacking a target that the left actually cared about. Video games were still just glorified toys as far as normies were concerned back then, and supposed video game "violence" could also be tied to gun violence. Which was, again, a good way to play both sides since conservatives and centrists were concerned with violent crime, as were even "civil rights" orgs, while also helping to pad out the lefty gun grabber impulse.

However, conservatives quickly figured out that the Clintons were absolutely not who they presented themselves to be, especially since Hillary is just not nearly as talented of an actor (or liar) as her husband. Bill Clinton was meant to appeal to blue collar voters since the Dems had alienated so many of those voters by pushing extreme forced integration policies and being way too tolerant of crime, to say nothing of how many of the 60's leftist movements had purity-spiraled themselves into pure insanity by the 70's.

But since the Clinton Dems largely failed to recapture the blue collar vote, they instead shifted over to middle-class normie libs. And these voters thought of the Clintons as the reasonable middle-ground between conservatives and leftists, as laughable as that is today.

It's easy to forget that although the seeds for today's insanity were planted long ago, normie libs didn't used to be nearly as batshit as they are today. They still thought of themselves as patriotic, many of them still made a point of identifying as Christian, and plenty of them were desperate to distance themselves from political correctness, to the point that Bill Maher's old show was called Politically Incorrect. Pointing out that Democrats were leading the charge against video games doesn't prove anything except that Democrats were trying to pander to both blue collar voters and normie lib voters, and their gambit was a failure for the former but a success for the latter.