Honeysuckle plants are easy to remove or kill, but when you have to remove thousands (some requiring you to lay on the ground and cut), I can assure you they are not easy to remove. That species is causing what will eventually require specific and direct government intervention (beyond EQIP) because of the damage it is causing to the environment. Deer will bed in any shrub and I'm sure would much prefer plum, dogwood, or hazelnut, etc. but those native species are displaced. I'm sure its browsed in some places, but it is minimal and not preferred. If it is being browsed heavily (I hear this repeatedly and have never witnessed it, must not like it too much or it wouldn't be invasive), your habitat is in poor shape. Its proliferation will also end oak regeneration in our forests.
People also ignore the decerease in land value it causes. Potential buyers are unlikely to desire BH monocultured landscapes, especially with no timber regen once logged. Call up a forester or other operator and get quotes for running a skidsteer and following up with a foliar spraying for two years. These add up quickly on a per acre basis.
Of course you're not going to kick deer up out of an empty pasture. That's a classic false choice argument. Bush Honeysuckle v open pasture, no one is advocating for the latter.
Remove all the honeysuckle, AO, and MFR on your property. Especially if you are planting shrubs and trees. MFR is the easiest to deal with and my preferred method is basal bark spraying in dormant season. BH and AO should be your initial focus. I prefer cutting the stump and using herbicide, or if smaller in size, foliar spraying in mid to late fall (if you can do this and not disturb deer) is really effective. Don't use tordon.
Aerial spraying is a promising method for control as it can be implemented via drone and done in the fall when everything else is dormant.
If you let BH or AO go unchecked, it will eventually (2-3 years) outpace your ability to control it. You will always be fighting it as one of your neighbors will let it go unchecked. I really need to write up a canned cohesive response for my anti-invasive rants. My experience with BH and AO is limited to the midwest, perhaps in other regions it is less of a problem.