According to some of the saner takes on the Pureblood-centric world, the emphasis on Purebloods not associating with Muggleborns is to make sure that the ability to perform magic will breed true in subsequent generations. Muggleborns marry into other recently Muggleborn lines until the rest of the Wizarding world is assured that these Muggleborn lines will continue to breed wizards and nothing but wizards, with no Muggles coming of the unions.
What, then, is the stigma attached to a house when the most Noble and Ancient bloodlines throw a Squib? Alliance-by-marriage seems, at least in fanon, to be a large part of wizarding culture. Can one magically analyze the genes to see whose family carried the (recessive?) weak magic gene? If not, one might imagine that no one would want an alliance with any offspring of Argus Filch's parents. I seem to recall Neville Longbottom mentioning that his family had worried that he might be a Squib for some time, and had been quite relieved when he proved to have magic after all. That relief might well be more than "Ah, the kid will be able to take care of himself, then," and could very much include "Good, now our family will not be excluded from the current generation's pool of marriageable witches and wizards."
How long does it take a wizarding family to ensure that their bloodlines are to be considered safe again? One would, of course, once a Squib showed up, be sure to pick the cream of the crop of the new Muggleborn witches and wizards to marry off to any siblings or cousins who do show magic... In that way, having a Squib in the family could be a blessing in disguise. One's family could snap up some of the newest, brightest talent, and regain strength a few generations down, when the unions stopped throwing Squibs -- and one could always blame the Muggle blood of the Muggleborn partners for those Squibs until they stopped appearing.
Since Harry was the son of a Muggleborn witch, would he have been a good pick as a match for an old-blood witch if not for the whole hero status? I somehow doubt it. The core of the Old Blood, the ones whose families haven't dropped a Squib in centuries and never will would already be sniffing and lecturing their daughters on the inadvisiblity of getting too close to the Boy Who Lived even so.