Why could injury or illness to a tendon affect the ability of the attached muscle to move a bone?

Attachment to bone. The muscle which is the engine of joint movement can only function through its attachment to the bone by way of the tendon. While there are many distinct types of tendons, some broad and others narrow, they are always the interface between muscle and bone.
They are connected. Muscles arise from one bone then blend into a tendon to attach to the next bone. This muscle tendon unit is the motor unit that makes joints move. If any part or insertion is injured it could effect the funtion of the whole muscle unit system.