Bedrockgames
I post in the voice of Christopher Walken
That's my point of disagreement. The only difference between the two is the number of choices on the menu. The fact that the menu is 100% derived by the DM and the players can only react based on whatever choices the DM hands them means that it's not really any different. Since, frankly in most sandboxes, most choices won't actually be on the table because of the level system - that vampire in the castle is off limits for that 1st level party, even if it is technically a choice.
Whether the menu comes from the GM doesnt matter. I don't know why you keep bringing this into the discussion. I mean it matters if what you want is for the players to have some control over that aspect of the setting and play, but it doesn't matter if we are just talking about whether players have freedom of choice in a sandbox. And this last point is simply untrue. A solid sandbox is going to have a wide variety choices available at multiple levels. It isn't going to be well there is this one dungeon you can service, then you level and can go to the next dungeon. That isn't how they are typically designed.
No, absolutely not. You are just asserting this is what a sandbox is. Anyone who has run them, knows the point isn't to just present the players with one choice, that you have already decided for the night.At the end of the day, it's just the players doing whatever the DM decides the players will be doing today.