I know you're thinking full suspension, but I'd recommend a hardtail 29er for what you describe. IMO, a hardtail 29er is undoubtedly the more useful of the two. $2K-$3K is going to get you either a base model FS bike or a pretty nice hardtail. You can use a hardtail as a gravel bike, it will climb better and it will be easier to maintain. Hardtails are just fine on the vast majority of trails, they'll certainly be up to what you describe. They're also the best way to learn to ride off road properly.