You didn't mention where this is, but I highly doubt it is accurate information.
It was the information given by those most likely to respond and make a decision on charges if there is a complaint. I believe the point being throwing the animal away makes it clear the purpose is to rid a pest and not "hunt". I can tell you in my experience ODNR really does not seem at all interested in dealing with city squirrels or rabbits instead pushing that off on local police.
As for the garden, once I knew it was squirrels I was dealing with, exclusion efforts were much more effective than trying to figure out a way to exterminate all the squirrels in my neighborhood. I wasn't expecting to find squirrels in my traps. As far as the legality of trapping nuisance animals within city limits generally speaking, I don't know, but my county will lend you a live trap for that purpose.
As far as your concerns about seasons and such, in cases where personal property is being destroyed by an animal I could care less, with the possible exception of an animal that is endangered. If I were to come home tonight and find a deer in my garage it would likely have a very limited lifespan, assuming it didn't quickly find its way outside. Out of season, clearly a valuable game animal, I don't even have a general hunting license let a lone a deer tag for this year. All irrelevant if it is stuck in my garage and kicking around tearing things up. The only difference would be I would call ODNR and ask them what they wanted to do with the carcass.
If it was quite calm I might wait to see if an ODNR officer wanted to come out and then see how they wanted to handle it. My guess is they would offer to shoot it or say they were not interested and to call the local police. I'm not going to push it around my garage and let it tear things up trying to get it out though. Not shooting it with a tranquilizer dart and seeing what damage it can do in the time before it goes out. If that is what they want to do I am going to send them a bill for any resulting damages and bring suit if they don't pay it. If it is kicking around damaging my property when I find it immediately gets put down. If ODNR wanted to pursue that case best of luck to them.
There was a case of a bowhunter living adjacent to a city park shooting a deer that was eating his flowers in my area. I don't think ODNR ended up pursuing charges in that case, although they did charge the man at first. There were several city ordinances that were violated, resulting fines, and, I believe, community service. He was a doctor, so probably had a pretty good lawyer, but I don't think much of anything came of it.
Several here have posted they relocated squirrels. I'm wondering what distances and whether that seemed effective. I have heard of very few wildlife relocation efforts that didn't end to the effect of "..and they beat us home." A possible consideration in the future if you all are having success. How do they react when released? Those i have had in live traps were quite aggressive. Often repeatedly attempting to attack my hand while I was carrying the trap. Not like a rabbit which is pretty easy to release.