the strip of grass *shrugs* could care less about.
what matters is...making sure water runs off the driveway, and getting water into ditches, and making sure ditches carry the water away.
the strip of grass, to me says the normal driving path for the tires is "rutted" and water does not flow off the driveway but down the driveway in the ruts. causing more maintenance issues for the driveway.
if you can get the correct slope / crown to the driveway, i would want that first. and if you keep the strip of grass after all said and done *shrugs* so be it.
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with above said, i have gravel, dirt, and grass driveway/lanes/trails throughout property, the one thing i hate is loose gravel and having to mow it. the mower deck gets dents, and i just tear up finishing mower blades from rocks. requiring more frequent sharping of blades.
on other hand have dirt lanes. and having the grass can be rather nice if it is wet/raining, i can drive up on the grass without making ruts in the dirt.