French drains, and it worked perfectly.

I wonder if a 6" trench run down either side of the walkway, and filled with drainage gravel wrapped in weed cloth might do the trick.

I'm also wondering if I should use a doubled-up benderboard edging material to elevate the gravel/sand bed for the flagstone so that the finished surface is a couple of inches higher than the surrounding soil.

I spent 32 years as a Painter and I can't believe I don't know any landscapers to ask...

Thanks for your idea Nay!

good drainage.

You're welcome!

There are other brands like this, too
http://www.multi-flow.com/Menu/french_drains.html

I put in a patio between a house and a garage, because the patio ran next to a driveway it was sometimes driven on while making a sharp turn around a corner of the house. 4 inches of sand over 6 inches of gravel didn't support even a little car traffic very well. You might want to check on crushed stone mix that can be compacted as the base.

on edit...as long as depressions don't develop along the tire tracks, it appears that the slope is adequate to move the water away, I've got something like that slope on my yard away from the house and I get similar amounts of rain per year. It also doesn't seem so steep that it would present a problem with the pavers wanting to slide downhill under the weight and traction of a vehicle.