Mister Honeysuckle has seen it all and believes the most cost effective way to remove a honeysuckle infestation in urban and surburban 'disturbed' areas (which most of it is), is to first remove all the smaller ones that you can manually with the honeysuckle popper, then saw off and kill the stumps of the larger ones. After that, just wait and see what happens next year.
The ground disturbance created by manually removing the small stumps unleashes the seed bank and you will get a lot of new vegetation, most good, some bad. One really good thing is new tree seedlings. The bad is usually garlic mustard which you will attack quickly as part of your restoration schedule.
After clearing an area you will need to go and weed out by hand the new honeysuckles that are going to grow. Early spring or late autumn are the best times to do this (in the north) because they are easily visible then. The first 2-3 years you will be pulling ones that grow from seeds already there, and also re-sprouts from an exposed root or two that you missed while popping.
After that you'll likely get new honeysuckles growing every year from seeds deposited by birds and rodents from the bushes on your neighbors property, (Hey, I didn't say it was easy did I) but you only need to go out every other year because 2 year olds usually pull out easily, and the numbers of new sprouts is greatly diminished.
Your honeysuckle patch may be an old stand, with most honeysuckles 20-30 ft. high with 8" diameter or larger trunks, all of which need to be sawn down and the stumps killed. The natural seed bank under these stands are severely depleted (except for honeysuckle seeds), so not much will naturally grow. In this case, its best to have some native plants and seeds ready to plant after removal.
I look at it like this: Bush honeysuckle are nothing but big weeds taking over your yard or woodland. I have never weeded my garden and thought: "Thats the end of those weeds forever", because I know they will be back. There is so many honeysuckle bushes around now that there is virtually no way they won't return to a cleared area.
That doesn't stop me though, I'm doing whats necessary to take care of my own precious little piece of the earth.
A capable operator can pop out most smaller honeysuckles up to 10' high quickly and easily, and the best thing about that is you remove the stumps. You only need to walk through an area where all the honeysuckle have been sawn off and soaked with herbicide to know what I mean. You, and your kids, will continually trip over all them small stumps. You cannot see them because the new vegetation thats going to grow hides them. They will last for many years too.