I've always liked the definition of the timeline from a book series Time Wars. Think of time like a raging river. Throw a small pebble into the river - the water diverts a little bit for a few inches, but a couple hundred yards down stream, there's no change. Little changes to the timestream are largely washed away. You need to make a MASSIVE change to see any real upset to later times.
As others have said, most of the changes Doc & Marty try to fix are to their own timelines. They're not making any changes large enough to affect time itself.
So let's look further down the timestream - What is actually changed by saving Clara's life? They'd named the gorge after her, and that's pretty much it. Some people in Marty and Doc's present remember that the gorge was named after "some teacher" who fell in, but she didn't become a cultural hero that inspired hundreds of people to go into teaching or anything.
She's no Edith Keeler, in other words.
The small changes in the time line are largely harmless - Twin Pines Mall to Lone Pines Mall, things like that. A new name for the gorge is a pretty small change.
If there's any reason for not wanting to bring her to the future, it'd likely be culture shock. There's no knowing what sort of mental effect being brought a century into the future would have on her, not to mention any potential diseases or infections she'd have no defense for. Her life in the future might be horrifying, painful, and short.
That is, until, just like in the first film, he just said "What the hell", and apparently everything was fine.