How long has the "no genetic downsides" aspect of their biology existed? It seems virtually impossible that their entire evolutionary cycle had this quality, how else could they have evolved from simpler organisms in the first place? Or are they artificially engineered this way?
As a primitive culture, assuming they had a hunter/gatherer phase, the need to interbreed with other groups would be necessary to prevent small tribes from spinning off into their own separate species. Unless their planet had several near total extinction events that kept culling these other tribes, eventually allowing just one tribe to quickly flourish across the planet, it seems more likely that only with advanced technology could they realistically remove the negative aspects of HABITUAL incest (1-2 generations of incest isn't really a problem) within a small subgroup of the population.
Humans have several biological mechanisms designed to prevent incest. Pheromones that drive sexual attraction typically "turn off" closely related individuals in most cases. Your aliens probably have a similar mechanism from when they needed more genetic diversity. This would have to be overcome for widespread incest relationships to flourish, I would think.
Socially marriage and children, at least amongst property owners, nobility, and the like, was a mechanisms to cement partnerships, alliances, and resource sharing agreements. Incest relationships won't drive these things, it would just ensure continuity of control over resources a family ALREADY has, so I suspect in an advanced society there would still be plenty of reasons to have outside relationships. So under most circumstances I don't think incest as a normal relationship event would develop.
BUT, if there was some unique quality/ability that had to be maintained (like the Targaryan dragon riding ability from GRRM's "A Song of Ice and Fire") then that WOULD drive incest in order to cultivate that genetic ability and limit it's spread. Perhaps certain technological devices were keyed to specific genetic sequences (and the ability to change those keyed sequences was then lost), so incest was used to ensure those sequences were passed on (as opposed to having to test all the offspring across several families, hoping the sequence pops up).
Of course institutionalized incest has other problems. Partners are necessary, but siblings are going to be staggered in age and gender. Even bringing in first cousins (and presumably the parents) you will have large gaps in age between partners and potentially surplus members of one of the genders. What to do with this surplus? Run them off to a celibate priesthood? Marry them off to outsiders (and risk losing control of the family genetics?). Seems like a source of rebellion to the ones with family partners. Of course if the family is genetically homogenous enough, perhaps they don't partner at all, just mate when and with whom they like since they may not feel a strong personal desire to parent a child so long as SOMEONE in the family does it (an extreme interpretation of the "it takes a village..." concept).