It seems like maybe they are a bit more stable if folks are getting in and out easily? Have you compared loading dogs into each type?
I'd really like to have the flexibility, but the sit inside kayaks seem like they might be able to handle non flat water a bit more easily? We might do some river kayaking, too.
You would need to get to rapids level of water before sit on top kayaks become unsafe in rough water.
I have been kayaking in the ocean, extremely choppy water with white cap waves and nearly 10' swells and haven't even come close to taking a dive.
The benefits of a sit inside are speed, not stability. It takes me 5 seconds after flipping a kayak to get back on. People I know who free dive off yaks get in and out of them 150 times a day.
I launch my kayak in the surf, and land it there too. It doesn't get much rougher than that.
The other advantage of sit inside is technique. You can do a bit more in them. Ie doing a roll.
But those advantages are irrelevant if you have a dog with you since it's not like you're going to the rapids with a dog. In a scenario you would bring a dog along, sit inside and sit on tops are mostly equal. On Sunday, I kayaked with syl and my niece (6 years old) on my yak with about 50lb of gear along the choppy beach in the afternoon.
Didn't even come close to tipping even with both of them moving from side to side, close to me and to the bow.
The best thing you can do is go to a demo day and try them out and decide for yourself. If you plan to do rapids, definitely get a sit inside. If you plan to do fishing or diving or camping, get a sit on top. If you don't plan to do any of those, pick whichever feels most comfortable.