If you've got the pitch, it should work.

What is the correct amount of pitch needed?  The surface of the driveway is only about 4" below the soil level under these downspouts.  This is over a span of about 18 ft.

If you've got excess water, definitely work to redirect some of it :-) I'm not sure, given our zone difference, if what I do to keep a wet zone (downspouts are an issue in one of the area I'm dealing with...) interesting and use the moist soil in that area would work for you or not but it might offer some ideas on how to use the moist soil if any is left after rerouting the downspouts.

I have planted a mix of Astilboides and Rodgersia where the neighbour's downspout directs water to our property, and attached a flexible pipe to the end of one of our downspout and ran the pipe under a path to exit in the midst of the plantings. There is also an Astilboides planting near the 'wet corner' area where rain runs down three slopes to drain into that location. How wet the soil is affects how big the plant leaves get - in drier times the plants are smaller. They send up tall astilbe-like flower stalks which I usually remove as it is the leaves that are the main attraction.

Some pictures - it's hard to see the downspouts in the first two but they are there if you look closely. The moisture in the area shown in the third picture is just from normal rainfall.

The neighbour's downspout at the corner of their house direct water to the right side of the plantings; the bed slopes to the left so the water runs down to bed.

This is the same bed but looking at it near where our downspout runs under the path (about where the honeysuckle stem is resting...) The extension pipe comes up in the plantings near where the neighbour's downspout also directs water.

This planting is near the 'wet corner' and all the water is from rainfall. In drier springs, these leaves can be half the size! We do not do any supplemental watering....

Some ideas to think about if you still end up with moist soil to cope with. I really like these plants but I have no idea whether it would work in your zone or not!

You could dig a sloped trench, line it with gravel, wrap some corrugated perforated plastic pipe with geotextile fabric and bury it. The water that didn't run out the sides of the pipe should come out the end.

I believe you are limiting yourself to the landscape bed between the house and walk to catch 3+the garage end downspout and route them to dump on top of the driveway.

Under any circumstance, you do not want to bury a perforated pipe that will deposit water next to the house foundation. And, you should not use corrugated pipe to begin with.

You really have to think bigger, by going under the walk and out to an area that is lower, or create one. You'll be getting water away from the house and not creating an ice hazard on the drive and entering the walk.

@Paul_DE-7a a French drain would be ideal once you figure out where to route it to (away from the house. You can bury a catch basin that will handle more. I had a situation that didn't have a drainage issue but did involve a flower bed. I dug a trench and connected a flexible corrugated pipe. I used a fake rock cap. I covered it with plants and mulch. Worked great no problem.