Decision tree with multiple outputs
You basically answered your question: if y1, y2, y3 are independent, there is no point to use a single model. If they are dependent (say, y1=1 implies that y2 is not 2), then the single model helps to exclude the non-existent case.

However, if y1 is a function of x1 and x2, and y2 is a function of x3 and x4, and x1,x2,x3, and x4 are independent, y1 and y2 are also independent.

I would start from whatever seems simpler and more logical. Based on your description of weak dependence between variables, multiple models seem more promising to me.