It's used on some public trails where I live. DG can also be tracked in, but it's probably no worse than sand. I'm guessing, but I suspect the landscapers you talked to have only used DG for a walkway. You have to put down at least 1 & 1/2" for a walkway. Then it is watered in and tamped down, because a walkway needs to be compacted. However, let's say you cover dirt with half that much and it only gets rained on. Compaction will be minimized unless you or your dogs travel regular paths. Even then, it should be possible to till DG into the soil.

One big caveat is what kind of soil do you have? Lighter soils that are mostly sand or silt won't be as affected, but clay soils tend to become hard and concrete-like when sand is added. Since DG is similar to sand it might do the same thing. Use this link to contact your local County Extension Office. Ask them what type of soil you have and if there could be negative effects to adding DG. If the office has no experience with adding DG to soil, they may not be able to give you a definitive answer, but they can probably tell you more about your soil in general.

If you do find a landscaper who agrees to do this job the way you want, I recommend you write out a contract detailing EXACTLY what you expect. Things that aren't written down can become points of contention. Good luck.