This!
Are they older by chance? Sometimes with the older generation, a handshake or verbal yes (regardless from who) is just as a good as a contract. Obviously that isn’t the case, but that is how they view it.
You have to live next to these people and it is easier to blame someone else to save face than have years of hateful neighbors.
I think it’s really kind of you to offer to pay half as well. you are being more than generous and i hope they realize that.