I feel like you're severely underestimating how much and how many ways dogs can be useful dogs can be useful in hunting.

Sure, they can give chase as persistence hunters, like you said

But they're also useful for tracking, let's say you shoot a deer, and unfortunately it's not a clean kill, and it runs off into the woods. You might spend hours, even days trying to find that deer, following tracks and blood trails that may stop in places and be impossible for a human to pick back up, but a scent hound can follow that trail, saving a lot of time and energy .

Or of course just sniffing out an animal in the first place.

They can be retrievers, if you shoot a duck over the water, you now need to find a way to retrieve it from the pond. If you don't have a boat, that basically leaves swimming out there, and if the weather is cold, that puts you at risk of hypothermia. Many retriever breeds are much better suited to cold water than we are and could swim out to retrieve them with no problem. Or if you shoot something in dense brush they may be better able to retrieve it than you would without having to bushwhack your way to where it went down.

Breeds like beagles are incredibly effective for small game, they can flush them out of hiding spots for their human to shoot, or to tree game

And dogs can be used to great effect driving deer, flushing them and chasing them towards the hunters. This was actually so effective that some places have laws on the books that you can't use dogs to hunt deer because back in the days when commercial hunting was more of a thing, it drove some deer populations to near extinction.

For some dangerous game like boar and feral swine, their role might be more to catch and hold the animal for the hunter to dispatch safely.

And that's not to mention the value they can have for herding and guarding livestock, especially if you're otherwise trying to go it alone or with only a small group of people, because you're likely going to have other tasks to see to besides just constantly watching over them. To some extent, donkeys can fill a livestock guardian role, but if you need to round up a flock of sheep, they're not going to be much help with that.

In any of those cases, they're not necessarily generating value on their own, but they can significantly improve how productive you can be with minimal effort on your part.

As for barking, that's a mixed bag. Yes, if you're trying to avoid detection, it can be a detriment. But in other cases it can be a deterrent, making intruders and predators wary of intruding into your territory knowing that there's a dog around, and it can give you an alert when there is an intruder.

And in some cases and environments, they can be useful as small draft animals, dogs can pull small carts and of course sleds.

Not every dog can fill every role of course (although some can certainly fill multiple roles) and they may not make sense for every scenario, but I think they're worth a lot more consideration than you're giving them.