Are there aortic valve repair surgeries that don't replace the valve with an artificial one?

Yes and no: The valve is replaced. But the choices are mechanical, "pig", bioprosthetic. The mechanical valve lasts a life time usally. It requires anticoagulation with warfarin. The pig valve is by in large being replaced by bioprosthetic. They will not last as long but wont require long term warfarin.These will last 10 to 15 yrs usually. If your young a mechanical valve that wont need re-do surgery.

Yes: There are several operations that in fact repair the aortic valve or preserve the native valve tissue while replacing the diseased aorta where the valve attaches.