The breeding program was indeed the culmination of millenia (according to the Dune Wikia more than 10,000 years) of careful breeding across multiple families and genetic lines. The final strands of this program were contained in the Atreides and Harkonnen families.
The "breach" referred to in the movie is that the plan called for Lady Jessica to give birth to a daughter, rather than a son, and that this daughter would then be bred to a Harkonnen male to tie the two lines together. Presumably this Harkonnen male would have been Feyd-Rautha Harkonnen, Baron Harkonnen's favored nephew, although it is implied that it did not have to be Feyd-Rautha. However, it was quite clear that Feyd-Rautha's genetic makeup was extremely important. Important enough that the Bene Gesserit sent Margot Fenning specifically to seduce him and become pregnant with his child, so that the line could be preserved should Paul Atreides kill him.
Presumably it was felt that there were important genotypes present on the Harkonnen Y chromosome that needed to be combined with an Atreides X chromosome, thus the imperative that an Atreides female breed with a Harkonnen male, to produce a male offspring (the Kwitsatz Haderach).
Such an offspring would have the full genetic components to become the Kwitsatz Haderach, and would only require the appropriate training (both in developing his powers, and in ensuring loyalty and obedience to the Bene Gesserit).
However, either by unexpected mutation, or simple miscalculation, Paul had all the genetic requirements expected from the next generation.
Regarding your secondary question, about reducing the danger of the Baron's influence through the genetic memory, I believe the breeding program would have had no impact on this. The Baron's genetic memory was able to exert control over Alia due to two factors.
Part of the Bene Gesserit breeding program involved allowing the subjects to unlock both male and female lines of genetic memory. Upon taking the Water of Life, a Bene Gesserit, if succesful, would normally unlock her genetic memory of all female ancestors, and thereby become a Reverend Mother. However, the breeding program allowed Alia (and later Ghanima) to unlock both male and female genetic memories. It was only by virtue of the breeding program that Alia was able to hear the voice of the memory of Baron Harkonnen in the first place.
The other, perhaps more important factor, is that Alia was raised to Reverend Mother awareness, along with the attendant opening of her genetic memories, before she was born. This means that she did not receive the training, or even basic development of personality, that allowed Reverend Mothers to interact with their genetic memories safely. This is why the Bene Gesserit felt that she was cursed, and extremely dangerous. They feared the risk of one of the personalities stored in her genetic memory influencing, or even utterly dominating, Alia, which is a phenomena known to them as Abomination. In the end, this fear was justified, and frankly it seems remarkable that Alia lasted as long as she did.