When some people talk about "Plymouth Rocks" they are specifically talking about Barred Rocks. When others talk about Plymouth Rocks they are talking about any of the eight recognized colors/patterns of Rocks. I just call them all Rocks to avoid the possible confusion. Are you using a different definition for Plymouth Rock?
I've had chicks from Cackle and other hatcheries, not in your breeds or hybrids though. My Cackle birds performed as well as any other hatchery bird. I don't think your issue is a Cackle problem.
In my opinion the issue is more of your expectations or maybe experience. Production can vary depending on their age, how long they have been laying relative to the molt, and how they are fed. Hot or cold weather can affect production, I get a drop in really hot weather and the shorter days and cold weather of winter can cause a drop not related to the molt. Mine go broody a lot, that cuts greatly into production. At least your current breeds/crosses shouldn't go broody much, probably not at all. Manipulating the lights can affect production, either for the good or bad. If they have been laying for a long time since a molt (or the onset of laying for pullets), their bodies can get worn out and production drops. Quality of the eggs may drop too. It usually takes more than a year of consistent laying for these effects to start showing up.
Another potential issue is that you don't have very many chickens. I don't know how many total you and your friends have but with your numbers just one chicken that isn't performing up to standard can have a fairly large effect on total numbers of eggs. Regardless of breed/cross some individual hens just lay better than others. They are all individuals. You have to have enough chickens for averages to mean something. With only six chickens covering three breeds/crosses, you don't.
I don't know enough about how you manage them, how long this has been going on, how you feed them, or their history of laying to be very specific as to what the problem is or how bad it is. Sometimes I can average 5 to 6 eggs a week from all my hens, others I'm lucky to get an average of 3 to 4 a week per hen.
I've had chicks from Cackle and other hatcheries, not in your breeds or hybrids though. My Cackle birds performed as well as any other hatchery bird. I don't think your issue is a Cackle problem.
In my opinion the issue is more of your expectations or maybe experience. Production can vary depending on their age, how long they have been laying relative to the molt, and how they are fed. Hot or cold weather can affect production, I get a drop in really hot weather and the shorter days and cold weather of winter can cause a drop not related to the molt. Mine go broody a lot, that cuts greatly into production. At least your current breeds/crosses shouldn't go broody much, probably not at all. Manipulating the lights can affect production, either for the good or bad. If they have been laying for a long time since a molt (or the onset of laying for pullets), their bodies can get worn out and production drops. Quality of the eggs may drop too. It usually takes more than a year of consistent laying for these effects to start showing up.
Another potential issue is that you don't have very many chickens. I don't know how many total you and your friends have but with your numbers just one chicken that isn't performing up to standard can have a fairly large effect on total numbers of eggs. Regardless of breed/cross some individual hens just lay better than others. They are all individuals. You have to have enough chickens for averages to mean something. With only six chickens covering three breeds/crosses, you don't.
I don't know enough about how you manage them, how long this has been going on, how you feed them, or their history of laying to be very specific as to what the problem is or how bad it is. Sometimes I can average 5 to 6 eggs a week from all my hens, others I'm lucky to get an average of 3 to 4 a week per hen.