How to get rid of brush and vines for good?

We recently moved into home on 10 wooded acres.

There’s a ton of brush in-between the trees. My family loves hiking so I’d like to clear some of this up. We’ve done some brush hogging, but it keeps growing back.

It mostly annoys me because it looks ugly and makes it hard to hike (some of the brush, not pictured, has pickers) but also the vines that get too out of control are killing trees.

Is there any way to groom our land in a way that prevents this garbage from growing back?

The photos attach show after a few months of brush hogging.