Eventually, the trait will breed true. Pets, like the common house cat and dog, are bred to be domesticated animals. They cannot survive in the wild like their distant cousins the bobcat or Dingo. Domesticated animals need humans to survive. Therefore, I do not think it is wrong for humans to have pets because it leads to their survival.
Most pets are like children to the owners.
Most of the time, pet owners spend a lot of time with their new pet just like they would a new child. As with putting pets in cages, I think it is perfectly acceptable. We put newborn babies in cribs to protect them. Likewise, we need to put our pets in cages to protect them also. It does not follow, though, that the pet should be in there all day.