Hi Mark, I'm dealing with a similar situation on my property, so here's what I've learned.
You can certainly do the "cut and paint" method, but just follow these tips: only do this when not frozen, and when not a drought, and immediately paint the herbicide onto the cut stem. So, this technique is actually best done early spring through mid fall.
The main advantage of cutting the plant down to the ground (if you don't simply paint the stem) is that the plant has to expend energy on sprouting (which weakens it) and you then only have to treat the sprouts instead of the large plant which you have just cut. The sprouts can then be treated with herbicide.
Digging up the honeysuckle is fine, but I'd recommend immediately tossing down some grass seed (like a conservation mix you get at a hardware store) and some shredded straw. This will help prevent the erosion. You could even replant the spot with natives
I like your idea of doing the project in stages to help protect the hillside. Maybe work cross-slope (along the contour) and take a pass or two, then wait 1 year before doing another pass. This allows vegetation to become reestablished before you remove more honeysuckle.
I've attached this Ohio-based factsheet on controlling honeysuckle. Pages 3-5 cover control methods, including the "cut and paint". The section on the herbicides will be especially important (since I'm in NY, I'm not familiar with Ohio's regulations on this).