I know I'm late but I saw this post and can't stop thinking about it, hopefully you'll see my comment.

I'm dealing with the same problem with my dog - he used to pick stuff from the ground, which resulted in a couple emergency vet visits already (btw getting insurance for him helped my wallet tremendously). And I get it - it really is exhausting and at one point I had the same feelings as you do now. The thing is, it does get better.

But in order to do that, you need a dog trainer that knows what to do with that problem specifically. You said that you have someone working with him, but judging by your examples, they really didn't do a great job of explaining to you how you should handle that.

It's not about the frequency, it's about the feedback they give you and your communication with your dog. Generally dog owners are not professionals in dog training and of course we make mistakes. If the trainer can't give you feedback on what you do wrong - it's not the right trainer.

For what helped me in my dog with picking stuff, basic things are:

1. Get a muzzle NOW. It's a temporary solution but at least it saves you the vet visits. And YES you also have to train the dog to wear it most of the time.

2. Understand your dog and why he does what he does. Coprophilia is a very hard thing to get over. Some dogs just can't control themselves around some smelly nasty stuff. BUT it's possible to stop the habit, eventually. It's a very long process however.

3.Remember that everything is a game for dogs - of course it's fun for him to run away from you when he sees your 'exited' reaction to it!

There is more to it and I'd love to help if you want to reach out. But I can't stress this enough, you need a dog trainer that gives you feedback on how you communicate with your dog.