Another possible option for killing weeds AND seeds: Use a roll(s) of heavy plastic (I used 6mil at 20' x 100' for a smaller area..).
- Area in adequate sunlight.
- Plastic sealed to keep the heat in.
- Re-use plastic for multiple sections of land.
- At least two weeks per section.
I like my scythe and goats too. I am no expert, but my Grandfather's old scythe has become one of my favorite yard/hobby-farm tools. I will acknowledge that it has taken effort to sharpen an old blade and work within limitations of old handle grips. However, I think a used scythe could be good to try - just to see if you like it before spending so much. (Remember too that a good weedeater could cost $300, still require gas and maintenance, and not last as long.)
You may be able to find a couple of goats for free or cheap...