The Cathari were the Albigensians. The Albigensians were an expression of Bogomilism (probably). At heart, they were yet another Christian gnostic sect. The basic premises that they (and most other Christian gnostics) followed were something like this:
Material reality is inherently evil.
The god who created material reality is either an inferior being (demiurge) or Satan, not actually God. The Albigensians believed that the OT God was Satan.
Since the Hebrews explicitly worshipped the creator being, Judaism is inherently evil.
Most human beings are, at best, mere animals. Soulless bodies.
Souls come from outside the universe and were lured and trapped here either by the creator being (if creator=Satan) or by another evil being (if creator=demiurge). These souls were trapped in the misery and ignorance of flesh.
Christ was sent by the true God to give Knowledge (Gnosis) of the real situation of the universe. He taught this secretly to only a chosen few who were then to seek out the true humans–the ones with souls, and help them free themselves.
Since fleshliness is ultimately inferior, or possibly downright evil, sexual reproduction is at the very least frowned upon–it only creates more prisons.
Albigensians had a “softcore” approach. Ordinary believers could live a less stringent life than utter perfection (only demanded of their leaders). When they died, they could expect reincarnation as humans if they were virtuous or as animals if they were not. The Perfecti (their leaders) were believed to become Christs.
More detail can be found in The Other God: Dualist Religions from Antiquity to the Cathar Heresy. Unfortunately, all the online information sites I’ve found have been extremely propagandistic, either for or against, and should not be trusted. This particular book tends to avoid that.
As you can see, since the Albigensians completely reject physicality, the idea of a “Holy Bloodline” that was sexually transmitted would have been positively blasphemous to them. Thus, claims of a “Sang Real” among the Albigensians rely upon a credulous audience ignorant of Albigensian doctrine.