I'm sure I've mentioned my own fanon about who the Cetra were, which is that being Cetra was more of a cultural difference than a genetic one. Sure, there were latent genes in the human genome (telepathy, empathy etc) which the Cetra peoples knew of and could probably activate through certain rituals or processes. Knowledge of these rituals or processes probably would have been kept by a certain caste, or possibly even a certain clan, within the Cetra culture.
I suspect the Cetra, as a nomadic culture, were already in decline when Jenova arrived on the Planet. There are two sites which show definite tendencies towards a more settled society (the City of the Ancients, and the Temple of the Ancients) had emerged within Cetra culture prior to this. The last remnants of the nomadic Cetra would probably have been the ones who were responsible for the sacred knowledge of how to activate the hidden skills and abilities within other people. They'd travel from group to group, assisting those who had the capability to develop the hidden talents, and maybe acting as roaming priests and healers for the various communities which were on their particular route. It seems likely Tifa's Master Zangan followed one of the last remnants of such a traditional lifestyle - travelling from town to town, teaching children all around the world what he knew, showing them how to activate Limits and similar (this explains why Tifa's final Limit Break is only available as a letter from her Master). The rest of the Cetra had chosen to assume a more sedentary lifestyle, and the old knowledge was slowly forgotten, to the point where it was just myth and fable, and nobody really believed things like that.
When the Jenova Crisis occurred, it seems likely the nomadic Cetra would have been the ones who'd be summoned by the more settled Cetra on the Northern Continent to help out. After all, they were the ones who were keeping the old knowledge alive. It also seems likely they were the ones first targeted by Jenova (there's a statement by Ifalna that Jenova infected "several camps" before the remaining Cetra managed to seal her). The destruction of what was effectively the Cetra knowledge base meant the loss of a lot of the aspects of being Cetra - and with the loss of the rituals or processes for activating latent abilities, these abilities would vanish out of the general population within a generation, going from being rare but unremarkable "gifts" to being extremely rare evidence of being "cursed" or "witchery".