What you are asking for is a world that is fundamentally classical, with limits to our understanding.
There exists a cooperative game for a team of three people. We ask them to go into rooms that will be completely isolated, and they won't be able to communicate with each other. They can come up with any strategy that they want to in advance. Once they are in their separate rooms, we will ask each of the three of them to hit buttons labeled 0 or 1 to make the sum of their three chosen numbers even or odd, we will collect the numbers and add them and determine if they win.
One fourth of the time, we run what's called a control experiment, where we ask them all to make the sum of their numbers even: and the team wins if their sum is even. The rest of the time, we choose one of the three at random to “betray” the other two: we ask the traitor to make the sum even, and we ask the other two to make the sum odd, and the team wins if the sum of their three chosen numbers is odd. So the two nontraitors know that there is a traitor but do not know who it is, while the traitor doesn't know they are operating at cross-purposes to their team.
In the classical world this puzzle can only be answered 75% correct: you have to choose one of the four situations to get wrong each time.
In the quantum world answering this puzzle depends on your ability to preserve and manipulate these delicate quantum states when you bring them into the communication-less room, but in principle nothing in physics stops a team from winning 100% of the time: it is purely an engineering challenge.
Somewhere within that space of what you don't know, in other words, can be hidden strange correlations that cannot be explained in the classical world. If you postulate hypothetical hidden information in this Heisenberg uncertainty then you have to also postulate that there is no way to truly make these rooms unable to communicate between each other, as the hidden information can travel instantaneously between two points no matter what stands between them. In our relativistic universe it must also be able to go backwards in time, if you go down that path. Most of us do not really like that approach, but it does have some adherents among the pilot wave enthusiasts.