I don't think that's the case. If it were then what would happen to humanity when the hero is killed and there is no other trained Belmont members? It would spell certain doom.
Ask Richter, he is a living proof that no other Belmont was there when needed, only Maria is shown searching him.
Although only a small few have ever combated evil I do feel that all members of the family receive some equivalent amount of training just in case.
I agree that maybe they are trained in a very basic manner, but that training being more intellectual than for combat. It would be the most foolish thing to train all Belmonts if they are not chosen carefully, since some of them could become evil, imagine a lot of Belmonts with grudge for not becoming the whip user being controlled by Dracula.
For instance; both the Lecarde and Morris family are of Belmont decent however they are not of the main family line. Yet they have training just in case they are called upon by destiny to confront evil. If something happens to the main family line then there must be a back-up contingency to thwart the evil.
Morris being descendants from Belmonts is right, but I've never heard about Lecarde being too.