Ancestral magic
Certain magical proclivities run in certain magical families, and it strengthens with each magical generation. As such, certain families (Black, Gaunt, Lestrange) tend to marry cousins to keep the blood "pure". The ancestral magic grows stronger but also more unstable as inbreeding becomes a problem. An extreme example is the Gaunts where they were basically squibs but could have "bursts" of uncontrolled magic. The Blacks hadn't gone that far yet, but they had a reputation for powerful magic and an unstable/explosive temper.
The Malfoys tended to marry other purebloods, but every few generations or so the married someone with some muggle ancestry to mitigate the inbreeding. The family was basically carefully run like breeding programme. Another example is Tom Riddle, who had access to extremely powerful ancestral magics because the Gaunts had inbred to the nth degree and became squibs, and then his muggle father "unlocked" the magic by mitigating the inbreeding just enough.
Even before the inbreeding has become as severe as the Gaunts (or even the Blacks) there will be an increasing number of squibs with each generation, despite having all-magical ancestors. These were left in the muggle world and eventually their descendants reentered the Wizarding World as Muggle-borns.
Magic is a dominant trait, but as such, it's not just inbreeding that leads to squibs, also the more muggle ancestry a wizard has, the higher the risk of the child becoming a squib.
As more and more wizards are descended from "new" magical families (or rather those who have disappeared into the muggle world for a few generations) more and more ancestral magics are dying out.
Purist supremacists don't understand how to live without ancestral magic and find it irresponsible to have kids with muggles, and want to ban both muggle relatives and muggleborns from the Wizarding World.
Others who aren't purist but pro-ancestral magic understands the catch 22 that blocked muggleborns off from ancestral magic: Ancestral magic is so important because it encompasses all they are, as such squibs are “feared” and muggleborns without ancestral magic are not considered “real” wizards
There are a few different factions in regard of ancestral magic:
General public – most don’t have ancestral magic because of muggle ancestry, and some have renounced it over the generations (Weasley, Prewett)
Fringe families – (not quite ancestral community) aren’t heavily involved in ancestral communities but do practice ancestral magic (Longbottom, Macmillan, Greengrass)
Conservative families – (ancestral community) an increasingly isolated group that is heavily immersed in ancestral magics (call themselves: Picts [see chapter 3]): typical Pict families + some outsiders who have joined throughout history: e.g. Bulstrode
“Sacred 28” – (ancestral community) more radical community who’s trying to preserve customs by rejecting muggle affiliations: Malfoy, Flint, Travers, Black
- > some are more radical than others, e.g.: Gaunt, Black, Lestrange
Families can move from one group to another – e.g. Bulstrode was “Sacred” in the 30’s but joined the conservatives, Weasley and Prewett were “Sacred” but gave up ancestral magic, Longbotton, Macmillan and Greengrass have become more fringe as they distance themselves from Malfoy, Yaxley, Black, etc.