Of course it does, if we accept business wants to minimise costs where practical, and we accept your assertion that "They have more control over how much they spend on labour than many of their other costs" then it follows that business would want to control the costs of that thing that they have more control over.
And if they do control the costs of labour, as you assert they can, then the cost of labour isn't going to be increasing at the same rate as their other input costs, which means that the cost of labour isn't what's driving their rising costs, at least not near the degree of everything else.