I plan on doing the southern half north bound next May & June. I will depart San Diego in early to mid May as a "get in shape" tour before starting the GD in Silver City, NM. I am like you in that I don't want technical stuff or too much punishment for going off pavement, just great scenery.
Go to CrazyGuyonaBike.com and look at the journals for the Great Divide. Most have lots of pictures to show you. The vast majority of the GD is on gravel or graded dirt roads forest service or county roads. However, some is single track. Occasionally you will have to push a bike due to steepness or roughness but I would guess that is less than 5%. About 10-15% is on pavement.
Note that if you go early in the summer, you may encounter snow drifts, rough roads that need to be graded from winter damage, higher water in creek crossings and/or fallen trees causing you to have to unload your bike to get over/around it. If you go later in the year, a lot more washboard, drier conditions (harder to find water), hotter, etc. but most of the above early problems will have been resolved. Pick your poison. I prefer wildflowers and a little cooler temps over dryness and heat. Plus the bears aren't as hungry supposedly
early in the summer if you stay away from their cubs.
Hope you enjoy the Divide!