Uh, wouldn't this have the opposite effect? The long winters would drive people to invent means of better preserving food, conserving heat in buildings, not to mention the large die off in population would create the need for labor saving devices. Nessesity is the mother of invention, and Westeros needs all the help it can get.
Sure, Westeros's need for technology is greater, but its ability to devote resources to technological advancement is near nonexistent. What capability for technological process Westeros
does have is pretty much all channeled into things that directly help with the climate issue, and in
that area they've actually made serious progress and are far in advance of the Middle Ages, Winterfell's greenhouses and the ability of the Maesters to calculate the seasons in advance when they have nothing but climate patterns to go on being the first few that come to mind.
But in other areas? Technological advancement requires that you have people with enough free time to either wonder about how the natural world works, or tinker with machines. There
might be a few of these at the Citadel, but the Citadel appears to be a deeply conservative institution that likely wouldn't support this kind of activity. Everyone else has no time to dedicate to anything but survival.
Finally, there's an overlooked issue: Westeros
does possess some unusually advanced "technology," but three hundred years ago, the major center of learning and technological advancement exploded, and apparently took a good portion of the laws of physics with it. All the time and effort that had been invested into such things as Valyrian steel and wildfire was rendered useless. So Westeros has actually suffered a major setback in its technological development recently, one from which it is just now recovering.