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...
A scythe has become one of my favorite trimming devices (second only to goats). Whatever the type, it must be sharp. Typically the blade requires touch-ups during use. (Minor comment - I didn't think not all blades should be 'peened'.)
In addition, to sharpness, proper technique helps tremendously (I am still learning). Even with my beginners technique, it mowes surprisingly fast.
The sound is much better than the powered variety.
Mine is extra special since it is one my Grandfather used! (It had been stored in the garages of my parents for 15+ years and then my own garage for another 15+ years until I actually tried to properly use it. I am so glad I kept and finally put it to use.