I wanted to stay out of this, but the enormous amount of wrong information is forcing me to respond.
1st-The dog warden did at one time do licenses for cats just like dogs. Over time the ability to enforce cat licenses got way too hard and it kinda got forgotten and ignored.
2nd-You can humanely kill a cat that is in your neighborhood if you want in the state of Ohio. Yes, some lawyer type can get on here and bring up many "gray" areas, but in general if the animal is yours, you can kill it humanely. A cat in your yard without tags or i.d. can easily be considered your cat.
3rd-The Toledo Area Humane Society does not normally handle stray cats. If you are a big donator to them, they can probably be convinced to help you, but not normally. They are also a non-profit organization. That does not mean they are "understaffed". The have a huge budget from donations and many, many gifts left from wills. Every employee is paid. The executive director not only gets a very good salary, but also gets a "company car". It used to be a fully loaded Jeep Grand Cherokee, but I don't know what it is now. Yes, they have many, many, great volunteers that help a great deal in the daily operation and special events.
4th-Humane Ohio is the organization that was instrumental in starting and running the Catch, Spay/Neuter, release program. The idea or basis of the program was to spay and neuter cats in a particular area so that they couldn't reproduce. The cats would still be occupying the area, so new cats couldn't move in, but you'd still have cats.
5th-I don't know where your yard is but 25-30 piles of cat crap in a week? I think you are exagerating a bit. Unless your yard is sand, you must have some reason those cats are staying around your yard. Besides, cats usually cover their poop. Are you sure it's cat poop? Maybe it's the canadian geese, or a small tribe of garden knomes. Try a motion detecting light, an automatic sprinkler, or one of those motion activated lawn decorations from harbor freight tools or other places. They will scare away any "feral" cat.
6th-I have a bowl of cat food outside for one major reason. The cats keep the rats away. Our wonderful neighbor's house was abandoned except for teenager parties which left the house wrecked. With all the trash and food lying around, rats were soon spotted in plain site during the day. The city said that they could not do anything about it because it was private property. They did say that if there was an alley, they could set traps in the alley, but not on private property. After trapping 8 rats, several even escaped the big rat traps, I chose to feed loose cats. The result, I get opossums, raccoons and cats, but no rats. I did some research on the opossums and raccoons and found that they are much better for my yard and much better in general than rats. I'll continue to feed the cats. I've been doing it for about a year now and have never seen or found cat poop in my yard.
Finally, while it is a choice to kill the cats, I can't see any normal human being doing so. Why is your yard the only yard that has become a cat poop magnet? Usually when I hear a complaint like yours, it is a stuffy old person who spends their time sitting at a window just waiting for a bird to land in a tree and poop on their car. Then they have an excuse to have the tree cut down. All the while complaining that they were "forced" to remove the tree because of the hassle it caused.
All kidding aside, if you want to give me an idea of your address, I can come by and give you some ideas and/or assess the situation. I used to work in the field.