This statement is false in my part of the world. Maybe different there but........
Burndown now and disc you might see new growth and spraying again in the fall might be effective.
However, taking you statement "Discing in the fall helps bring the dormant seeds up, so the spring application has a second chance to catch new growth". It doesn't work that way. Anything that might grow now will freeze and be done. Yes the roots are alive in the soil but there is no top foliage next spring. Horsenettle really doesn't show in fields until well into June. Way past the point of putting in a new seeding which for here the cutoff is really early May. Your area I'm sure differs on optimal dates. Even if the horsenettle was visible and you sprayed there would be plant back restrictions unless using Roundup only. There are pre emergent sprays that would reduce horsnettle before it starts growing but have no idea if any of them allows you to reseed immediately.
Not trying to start an argument but something doesn't seem right with what you want to do with regards to Horsenettle control.
As for the toxicity, there are a lot of plants that are toxic. Would take a lot of it to be toxic. On top of that unless you were force feeding it to them they are going to avoid it. Properly timed control spraying, managed mowing intervals, field maintenance, proper cutting heights are always going to be your best methods of control. You will not rid yourself of it. And mother nature will always persevere on your best intentions.