One question I have is, what kind of ground do you have underneath the topsoil?
@Kbar has excellent suggestions, I believe he is a heavy equipment operator and would have good experience in this field!
To me when I look at the pictures (that sometimes can be deceiving) it appears that the ground elevation (grassy areas) actually slopes towards the driveway! It’s one thing to have a low driveway but if the water runoff is sloped towards the drive to start with I would think that you might have a bigger challenge, not saying it can’t be corrected, just more challenging!
I built a driveway years ago and used limestone, i dug down a good 8” to 10” and layed down fist sized ( 2”- 3”) limestone 5”to 6” deep, packed that and let it settle for a week or so the used 3/4 minus limestone on top of that, having a small trucking company I needed a driveway that would support a loaded semi so that’s the route I took, I have a sandy soil about 1’ under the ground level so I have excellent drainage to start with so that helps a lot, that’s why my first question was “what’s underneath”.
If you go this route and have a quarry somewhat close, chances are they have trucking companies that haul their materiels for them that could bring in the rock, ask if they can spread the material for you, meaning as they dump they idle the truck ahead with the box up in the air with the tailgate chained to only allow it to open so far and spread the material, but it does take a experienced driver to do this, a lot less tractor work for you. Spreading larger rock can be somewhat challenging and some trucking companies might not do that, depends on there experience, getting the 3/4 minus material spread should not be a problem for an experienced driver to do, obviously you have to make sure of overhead obstacles, power lines, trees etc while spreading with the dump box in the air, from the pics that doesn’t look like much of an issue.
Like Kbar mentioned, that driveway has to be a bit higher so figure that into how far down you have to remove existing material.
I live in Minnesota so I understand the freeze/thaw cycle you have in mind!
Good luck with your project and let us know!
I did all my work with an older John Deere 2550 with a loader and a rear blade.