Stan: I believe that was a theoretical concern with the soft lead cube shot available in the 70s - 80s. Orvis promoted a Woodcock load and the box was marked "Not for covey shooting".
I have not heard the issue mentioned with the various plastic insert spreader loads. My limited experience was with Polywad (disc and post) Spreaders in a tightly choked 16g Smith. The pattern was a bit wider and I don't recall fliers, but they could have been off the 36X36 paper.
I never tried the cube shot. Can't remember who it was, but someone promoted flattened "disc" shot. I did pattern some of those. Pattern basically ceased to exist beyond 20 yards. Never used any on either clays or birds. I would not have used them around dogs.
I haven't tried all the spreaders out there. But I have observed that the Polywads--either factory or reloads using their inserts--spread much wider than the RST's. Enough wider that they might be a danger to a dog? All depends on how far away the dog is and how low you're shooting. The guns I shoot now at grouse, woodcock, and quail--often requiring close shots where you want a lot of spread--are open enough (cylinder or just a few thou constriction) that I don't use spreaders any more.
