You're entirely correct, I just wish to add my two cents so apologies if this comes off as my correcting you. Not only are they usually uninformed, it's just cheaper overall to neuter and assume the future forever home are just going to drop them off suddenly and leave them to it. Sadly, it's become a cultural thing it feels like so there won't be any other alternatives to fixing animals.
As for the question, you may be able to use Facebook groups or similar social media groups as people may post that they have a bunch of puppies they want adopted. Some are moderated to prevent people from making puppy mills for profit. There's also a chance, not sure if you're from a small town, that there may be a grocery store or gas station in town with a corkboard for various adverts or of course what's considered the towns advertising pole. I call it that as I've seen telephone poles with a bunch of stuff all over them advertising many things across my travels