Private health insurance can get you shorter wait times for elective procedures, and this can make a huge difference if your condition is affecting your quality of life.
Getting in sooner can also have health implications. For example a clogged artery is extremely serious, but not sufficiently life-threatening to qualify as an emergency.
Just be aware that not all insurance companies cover all surgeries. For example, you'll probably need a top-tier policy if you want to have major heart surgery covered.