You sound like you have some knowhow about dogs already. These challenging cases can force us to look at things from new perspectives. So I'm going to throw some ideas out there for you to consider. I apologize in advance if it's way off base, it is difficult to consult on dog training when you don't see the dog first hand.

I'm not a purely positive trainer, by any means, but something that really makes sense comes from that camp. They say that making pooping uncomfortable or scary to do in the presence of a human can be confusing to a sensitive dog. So if the dog has to go and thinks that it will be 'puppy shamed' as you described, it will not want to come to you when it is about to poop. Instead it goes away to a discreet location in the house where it can get away with the act of pooping. And we want them to come to us, be close to us, and let us know they have to poop. As you said, your doggo is non vocal, hasn't learned how to do that. There are subtle 'i have to poop' signs unique to each dog, you can observe it in a change in their behavior. For example a certain antsy-ness. If you dog feels very comfortable with a full feeling around you, you can start to observe that tell-tale behavior and predict when he has to poop. So that's one argument/mechanism for why 'puppy shaming' might be working against you.

But maybe that isn't the case with your dog. Hard to tell over the internet, haha.

Here's another thing to think about. Exercise and reliving oneself is very related. I first heard this one from a YouTuber named Stony Dennis. A dog will want to pee and poop before and/or after intense exercise. If you make the exercise/ feeding/ watering times very very scheduled, I.e. at the exact same time every day, his pooping peeing will become also more scheduled more predictable.

Also you aren't free feeding are you?