I think a band saw would be perfect.
I hope you have one.
You mentioned you are making folding picnic tables. I imagine that means you've got a hole for a dowel at the center of this rounded end to make a hinge joint with another board.
You need to make a jig to do this easily on your band saw. Here's what you need:
First you need a band saw blade that has maybe 3-4 teeth per inch and is no more than 1/4" wide.
You need a sheet of plywood to cover your bandsaw table that has a dowel sticking up about an inch. The dowel should be the same diameter as your hinge dowels, but it would be nice to have a metal one for the jig. The plywood should be anchored to the bandsaw table. The dowel should be located so that it's center is 1 3/4" to the side of the bandsaw blade's teeth.
All of your 2x4s need to be pre-drilled for your dowel hole and the holes need to be centered on the board. If off-center, you'll have problems.
Then, just drop a 2x4 on to the dowel and swing the board around 180 degrees and you've got a perfect half circle end that's concentric with the drilled hole.