I know the OP asked about the quality of honeysuckle as a nectar plant, but I'll share my experience of removing it from my property. So far, so good. But, it is a constant process due to reseeding by birds.
Two approaches work well for me. In the spring, roots are looser in the soil so it is somewhat easy to pull the living plant out of the soil by hand/shovel, or to use a chain and a tractor (even a lawn tractor can work) to yank the plants out. The rest of the year, cutting the plants down and applying glyphosphate in the form of Aquaneat (very concentrated glyphosphate) kills the stumps in one application, if the application takes place right after the cutting. The exposed cambium brings the herbicide right down to the roots just like it would if the glyphosphate were absorbed through the leaves. The dead material is easily removed the following spring.
Removing honeysuckle reminds me of that scene from Fantasia, where chopping the broom into little pieces just makes more little brooms. Thankfully, the smaller plants that might emerge the following year are very easily removed. Resprouting doesn't happen with the use of glyphosphate. New (or missed) plants are easily spotted in the spring and easy to remove by weeding.
The process of cutting and using aquaneat also works well with tree stumps, for what it's worth. Residential-strength Roundup just isn't strong enough to prevent resprouting of these.
Pete