Hi!
As you can see, when you put keyframes on your constrained objects, the channels turn green. This means they are weight blended. Now you can theoretically key that weightblending on and off with an attribute that gets created on the child object, but my advice would be to avoid keying constrained objects. Or to put it differently, avoid constraining objects you want to key later.
With your example this is acutally quite easy. because your rig seems to be fk only, you can just work with hierarchy.
Simply parent your objects to each other so they follow correctly (legs to hips, torso to hips, shoulders to torso, upperarm to shoulder and so on). This will give you a fully functionig rig that behaves like you want and you can easily key.
Now with more complex rigs you can avoid this problem by using offset/constraint groups. Simply group the object you want to constrain to a group that is in the same world position as your object and then constrain the group instead of the object. this keeps the channels from conflicting and blending values since the constraint is on the parent object and the keyframes on the child object.
I hope it helps!
