[mathilda]

I have a Yorkie mix who does not behave like a dog. I got her 2 years ago at 6 months. Her distant, non-engaging behavior has always been there and nothing I have done has changed it.
We go to the dog park 3 or 4 days a week. She does not go up to other dogs, she will not run with them or chase them. If one tries to engage her, she ignores them until they lose interest and find a better playmate.

as long she's healthy there nothing wrong with a dog that is not so socialble, not every dog is a social butterfly. they're still dogs.

Not once has she chased a ball; if one is thrown she barely glances at it. Toys of any type do not interest her.

toys are actually not natural for a dog, so not being interested with toys, especially, when they're adult is not that rare.
doesn't make them less of a dog.

The only interaction she gets excited about is small children. But since I don't have any and they aren't allowed at the dog park, that is only on rare occasions.

this sounds like a good dog. being people oriented is awesome. perhaps when you have the possibility to see how the children she likes act around her, you know what kind of behaviour make her excited. you can play with this way.

She comes when called and stays. Heel isn't necessary because she always walks on my left side without running off, most of the time a leash isn't necessary. But these behaviors aren't anything she learned in class. I have taken her for training with 3 different trainers and they've tried to get her to fetch and other actions that would lead to playing. Problem is, she doesn't want treats, so there is no way to entice or reward behavior.

this sounds like a great dog. a real dog. that just wants to be with his human. Dogs are not robots. they have personality and your dog is obviously a very nice, well-behaved young dog lady that like to interact with humans, but finds toys and treats boring....or the toys and treats were just the wrong ones. I yet have to meet a dog that isn't interested food at all...You sure she wasn't just full? feeding her by hand over the day for work and not from a food bowl can help. human food (in small doses) is loved by our dogs. sancho likes cooked potatoe, cheese, sausage, bread...if the pieces are wery small it isn't that bad for them, I just want it to work and not be totally poisonous.
did you try playing with her without a toy? some dogs like (controlled!) scuffling with their owner for example.

Frankly, all she does at home is lie there, she's not a companion, she's a fixture. This is not what I wanted from a dog and am thinking of getting rid of her and finding a real dog.

lying around and at home and accompany a human is wat a companion dog usually does. there's nowhere written in the job description that they have to love playing fetch.
having a ball junkie as a dog is just sad for the dog anyway.

Any suggestions?

delete these hollywood dreams of a "real dog" from your mind and look at your real dog.
it sounds like she's a calm, obedient dog, that is not obsessive about stuff, likes people and children and is not interested in strangers training her, just because they wave with some random dry treat.
This sounds like a genuinely awesome dog.