What does ketone testing do that blood sugar testing doesn't?

What Ketone testing does is, as you rightly surmise, proves whether you have ketones.

Essentially it's a measure of how dangerous your high blood sugar levels actually are. If you have high levels of ketones, you are likely to be closer to Diabetic Ketoacidosis and therefore at risk of ending up in A&E. Not everyone who has high blood glucose levels does correct it in the way you describe, so it's really a safety net to confirm your current risk level.