If I were to go w/the latter, is it as simple as dumping some gravel on the
lawn

No, you'll wind up with rocky mud every time it rains. Do it right, dig out
about 4" of the dirt and use it to fill in low spots on your lawn. Then get a
load of unwashed crushed white rock and fill in the area. The limestone fines
in the gravel set up almost like concrete after some time and make a very
sturdy driveway. Our local contractors use it a lot in road building. HTH

Joe