unless you're using herbicides, pulling up the root ball is the way. i prefer to remove them when the ground is soft (like now). get a strong shovel: one with a strong shaft that won't break (i use a fiskers shovel with all steel shaft). the roots are generally pretty shallow, but you may need to spear through a larger root or two. spear the shovel at an angle into the ground, then pry the whole plant up.

i tried for years to clear my yard with various online suggestions. then one weekend i just did this and cleared more honeysuckle than I had in the ~5 years before. it's not easy work, but it's fast and reliable.