Of course by machine a true polish or compound is the way to go.
If by hand, Scratch-X is probably the most user-friendly. The fact that it contains fillers certainly helps people think it's more effective than it actually is. The key to Scratch-X is repeated application and removal. Most people use it once and say it's not that good. Try it 3-5 times and the results should improve.
You will never be able to fully remove a bird dropping stain without taking off a lot of clear/paint. You are going to have to get it to a "satisfactory level" and live with it.
