Death is the default condition of all life. Neither you, your neighbor, little less the coons she befriends will escape the death sentence we all receive at birth..
Here is the dilemma:
Is it better to kill a few local 'hick' coons to pare the population down to what the ecosystem can feed without the coons becoming dependent on your chickens or your neighbors' kindness for food?
Or is it better to double, then tipple, and double and finally triple again the bear, mountain lion, and wild wolf populations so these apex predators can act as a check on coons, lonely old women, joggers, and 8 year old children playing too close to the woods?
Playing the who or what was here first game is a lame attempt to arrive at what ever outcome that you've already decided you want. Where does this start or stop, with T-Rexes, giant ground sloths, terrordactyls or passenger pigeons?
Coons, especially on the East Coast are lousy with rabies. The plan that the government came up with called for carpet bombing a wide strip from the South East Atlantic coast up through New England with baits that contained an oral rabies vaccine. Then each year the treated area would shift further East as rabies in coons was wiped out in the Appalachian Mountain area and each year the treated area would keep shifting further East until (I guess) all the rabid coons had been driven into the Atlantic ocean. Sounds good doesn't it?
The reality is that the treated or beaten area keeps moving further West each year in a vain attempt to stop rabies in the coons that made an end run around the vaccine program. I believe that Ohio is now in the bomb sight. As we move wildlife around willy neile with no plans or good reasons, your doing more harm than good.
You may well be moving a rabid coon into an area where rabies in coons is unknown. Or you could just as well be re-introducing rabies back into an area where rabies was eliminated. Don't, don't, don't relocate wildlife on your own.