My house sits on 2 acres, with 1.5 of that wooded. When we moved in 20 years ago the wooded section was full of bush honeysuckle. Figured out what it was about 8 years in and some winters spent some time really hacking it back and treating the stumps with gly like you suggested (straight from the jug, ~40%). That works pretty good, but I'd have to be really vigilant to totally eliminate it. The whole surrounding area has little woodlots that are choked with it, so I'm never going to eliminate it, but by whacking it back every couple or three years, I can keep it to small patches here and there.
Also had about 30 mature ash trees that are now, of course, gone due to the ash borer. That really opened up the canopy so the last couple of years I've been working harder on the honeysuckle and seeing what shows up with all of the new sunlight. I'm getting a bunch of spice bush, but also have some young oaks coming on, some crabapples, viburnum, hawthorn, mulberries, and the typical maple, sassafrass, etc. Just about all of that was probably there all along, just waiting for a chance; the rest the birds brought in. Hated to see those 24"+ dbh Ash trees go, but from a diversity perspective on my little postage stamp, it has actually worked out pretty well.
Windy way of saying that if you really work on getting rid of bush honeysuckle plus open the canopy some - then just keep an eye on what shows up, you might not have to do much else.