Agree with not wanting to use glycophosphate. Stuff grows back soon anyhow, and many weeds (which will take over after the grass is permanently killed) are at least partially immune to it. And I can't see string-trimming "miles of fencerows." Life is too short.
There are no "easy" solutions but here are a few:
1. Plant a sturdy groundcover like creeping phlox along your fence lines. You'd need to do this in early spring with seeds. Or, plant low growing bushes: berries, forsythia, etc., which will spread. Ornamental grasses also work well as they will spread also. Also you can cut them back in winter and have access to your fences for maintenance.
2. Longer-term, plant rows of trees along the fence rows. Grass/weeds like sun.
3. Shorter-term, in early spring while the ground is still bare, add at least 5 inches of mulch. (If the grass has already grown up, you can weed-wack the grass down to the ground and then put down the mulch; works for us when we've been too busy to stay on top of weeding.) If you want to implement #1 or #2 longer term, the mulch will help keep the grass at bay while your plantings grow.
4. Baking soda will kill everything for a couple of months. Better than glycophosphate but still puts sodium into the soil.
5. Let it grow and then once a year, just before the relatives arrive for a visit, trim it down.
6. Hire a mowing company to maintain it.
We are also getting ready to implement some extensive horse fencing, and I'm aware of the problem (love everything tidy, but hate the hours of tedious work, did I mention life is too short?). I will probably opt for #6 while I wait for #1 and #2 to grow in.
Let's face it, "miles of wooden horse fencing" were easy back in the day when everyone had a couple of gardeners/stablemen to take care of it all.....
Will be curious to see what you decide.