Yeah, using a sythe to "mow" the lawn is rough. It does need to be razor-sharp, and requires a lot of control/experience to do it well. And it will be very hard to get a nice, smooth, even, short cut on grass. I use one all the time for grain and weeds/grasses. The "stubble" left behind from normal cutting still is 2 to 3 times as long as a household lawn.
The closest thing to get a "better" cut on small sections of grass is usually called a "weed whip". It as a blade around 8 to 12 or 14 inches long, and is angled out from the handle to be parallel to the ground when you are swining it. But it still requires a very sharp edge, and only cuts less than 1 square foot of grass/weeds at a time. That translates to a whole lot of swinging it to do a household lawn. Plus, it still takes lots of practice to get that nice/smooth finished lawn look.
One suggestion would be to check out the local antique/junque stores for an old push reel-type lawn mower/cutter. The wheels drive the rotating cutting blades, so you pushing it around the yard powers the cutting. Most of the later versions were designed to be self-sharpening, but it doesn't take much to sharpen those blades. Altho they can be a bit ackward to sharpen. Sometimes you can pick one up pretty cheap. And new ones are still being made. They are also now considered the ... green ... alternative to a gas-powered lawn mower. But it ends up being a whole lot more work to mow your lawn with one - because you have to push it so much harder, and it cuts a smaller area each time. And they do not do well on "rough" lawns.
Using sheep to "graze" your lawn can work, but then you have to put up with all the other "adjustments" necessary to have them nip you lawn.
Just a few humble thoughts to share. Take them as such.
Mikey - that grumpy ol' German blacksmith out in the Hinterlands
p.s. Yes, I've used all of the above over the years. The "sheep" method was the worst. I spent more time building/repairing fences, chasing escapee's, and cleaning up "natural fertilizer" just to be able to use the lawn than it was worth. But they did taste good!