Good point, but it's in the state of Missouri, which DOES have a few things to say.
https://agriculture.mo.gov/animals/rules.php
This page appears to be part of the official state website, and has a link to "dog and cat law."
I followed some of the links from there, and this one looks very relevant:
https://revisor.mo.gov/main/OneSection.aspx?section=273.020
"In every case where sheep or other domestic animals are killed or maimed by dogs, the owner of such animals may recover against the owner or keeper of such dog or dogs the full amount of damages and the owner shall forthwith kill such dog or dogs; and for every day he shall refuse or neglect to do so, after notice, he shall pay and forfeit the sum of one dollar, and it shall be lawful for any person to kill such dog or dogs"
The page also has a note: "1957) Chickens held "animals" within the statute and evidence was sufficient to support judgment. Tillery v. Cook (A.), 297 S.W.2d 9."
I found several of the other sections relevant as well, such as
https://revisor.mo.gov/main/OneSection.aspx?section=273.033
That appears to give the right to kill a dog if it is on your property, and you have previously complained to the sheriff a certain number of times, even if the dog is not engaged in killing animals or injuring people at that instant.
I would say there is no "leash law" as such, but that dogs in the entire state of Missouri are NOT allowed to go around killing livestock or threatening people, and the law provides for killing the dogs and making the owner pay damages for the dead livestock. (I'm not a lawyer, so I might have read things wrong-- which is why I listed where I found it all, so anyone can easily go check it for themself.)
Right, apparently it's okay for them be at large as long as they don't cause any trouble. Which seems pretty unrealistic, dogs running amok, especially if they pack up, are going to cause all sorts of problems. I can understand not being able to afford funding for animal control services, but IMO leash laws are a must!