As you state the drain body has to follow the driveway, but the pipe connection can go anywhere in the drain. Therefore the drain pipe gets attached at the lowest point and then the drain pipes gets directed to wherever you have enough drop to flow to drain.

Well, once the water gets to the end of the channel, and hits the drain pipe, it can drop anywhere. But in order for the water to flow to the end of the channel, the channel itself needs to be slightly sloped. My question is that if the driveway isn't sloped in that direction, but the channel is, how to you keep the grate over the channel level with the driveway around it? Yes, I guess the channel doesn't really need to be sloped to get rid of a lot of water, but then you'll end up with a permanent puddle in there.

Any concrete flatwork should at least have a wash to it (even if unintended), so sloped towards some direction. A trench drain, or any pipe for that matter, can be perfectly level yet still drain water. Because at some point while seeking level the water will rise enough to overcome it's edge surface tension and pour out a lower outlet. And, AFAIK, any channel profile will have a lip below an end cap outlet hole anyway.

It could be possible to shim the grate at a slope to match the concrete but not the channel, but with a driveway and vehicular traffic it probably isn't a good idea.

Also, you would have to sawcut to install the channel into an existing driveway. I'm sure there is a scenario where you cut a bit beyond the channel footprint, and infill concrete which slopes towards the drain to alleviate what you are trying to accomplish.

Change the slope of the driveway please.

Saw cut drive,excavate a trench with bottom sloped in direction of desired flow, line trench with reinforced concrete while forming a ledge for grate below drive surface. It's a minor trip hazard but open concrete swales are often used instead of grate covered channel.