I recommend doing some instrument lessons if you haven't already. When I got my instrument rating I questioned whether the private requirements are actually enough. The skills that the instrument rating teaches in terms of preparation, workload management, and emergency weather scenarios make me question whether I was really ready to fly before I had it.
With the private or sport license you'll be fine in the majority of cases. I think my comment comes more from the edge and corner cases that more skills and experience help with, not your ability to work in the system at all.