How about a drain across the drive way just before the garage ?
Open grate style.
That really would not alleviate any water accumulation inside on the flatwork. It really needs to be sheet drained or sloped to an internal drain.
@OP- another thing to consider is your location and will the water freeze. Up here in the High Country, we no longer specify or design sheet draining as you are doing because of water accumulation at the door which will freeze and build up over time (even with a heated garage.) We predominantly instruct our plumbing engineers to incorporate an internal, centrally located floor drain with heat trace.