You have a good sense of how we use the word "as" in this context in your thinking that it means "while." Another way you could translate might be "since I have gotten older." The word "because," though, implies causation, meaning that getting older is what caused you to become more confident. If getting older causes one to become more confident, that is fine, but there might be a situation in which you have correlation and not causation, in which case, "because" might not be a good substitute word. For example, I could say, "As I have gotten older, I have become thinner." Did getting older cause me to become thinner? Not necessarily. So because would not work.
No matter how you analyze these sentences, though, the notion that an action is completed and does not continue into the present is a function of the verbs used, not the word "as." In the regular past tense, "I got older," the action is usually assumed to be complete. When you use "I have gotten older," the action continues into the present. That is called the present perfect tense. When you say, "I had gotten older," you are using the past perfect tense, in which the action is clearly completed and in a more distant past than when you use the regular past tense.