I'll break down your question into two.
I agree with you. It should be “must investigate”, not “may”. I'm not sure why there is a discretionary feature to this. The whole point of the standard is to reach a standard such that a power is triggered.
With respect to the disjunctive routes, I think it makes sense that there be a disjunctive route at this stage of the process because there can be cases where it's in the interests of justice to review it, where on the face of it, it may not appear to be a miscarriage of justice.
This comes back to what we call the catch-22 in post-conviction review. Post-conviction review often relies on new matters of significance, but persons in custody don't have the ability to investigate those matters of significance.
There may be an aspect of the case that cries out for a response, but you can't reach the threshold of reasonable grounds to believe because you don't have access to the investigative powers, but through the process of the interests of justice avenue to get to the investigation, you can access those resources such that you might eventually get to the point with new matters of significance that achieves the ultimate result.
It's a separate avenue that I perceive to be valuable for those who are in the catch-22, as we've called it, in post-conviction review.