Grass is the "sturdiest" groundcover there is.

Most groundcover that is rated as "steppable" means that setting a foot down on it with some pressure will not kill it.
But that is not what dogs do. Dogs run and shove off with their back feet, they corner and pivot and slide which scrapes the ground. They paw at interesting bugs and smells. They kick up an enormous amount of stuff just by running around being dogs.
So to even establish a ground cover one would need to fence off the area to allow the plants to establish an anchoring root system, and that establishment would take at least one full season/year.
If you are prepared to fence off the area for a year and not allow the dogs in, I too would wonder why your choice would not be turfgrass as it is by far the most durable thing you could use. It establishes far more quickly and regenerates more in one good week than groundcover could in a year.

Now, if we are talking Chihuahuas you might be able to get by with something else, but for dogs of any size at all ornamental groundcovers would be a maintenance nightmare and never look as good as a basic lawn.

I've maintained lovely gardens with enormous dogs for decades and the only way I have done it is by working with them and not against them. You can't fight their behaving like dogs and you certainly can't try to police them and prohibit them from enjoying their time outdoors. I do use a lot of temporary fencing when establishing new plants, but groundcovers are not something I would ever try to use in a dog zone except around the base of permanent objects which the dogs have to avoid anyway.

Okay, so I know this now becomes a bit off topic as we're in the groundcovers forum, but which cool season grass type is the best for dogs?

I would say that the best grass for dogs is the one that grows most successfully in your area. As you don't list a location, I'd say call your cooperative extension office or maybe a feed and seed store and see what is most popular.

I'd use wood chips.

The only way to grow things where dogs run is to have a sufficiently large area that the damage is distributed enough to allow for recovery. In a small area like you're describing, your best bet is gravel or wood chips.