Hi John,
The answer to this question in part is a "it depends" response. It depends on the skill or competency level the students need to achieve for next level success. I am not a good test taker so I would welcome the opportunity to demonstrate my competence by doing a project or assignment. But, I should do it by demonstrating that I know the material in a way that gives me the knowledge base that I need to more forward with my courses or career. I think that extra credit needs to be evaluated on a course by course basis before a decision is made about whether to grant it or not. As you mention this can open a door that can be difficult to close if one isn't careful in planning how it will be earned.
Gary