Proving statistically hiding for a simple commitment scheme in the ROM
Statistical hiding for a commitment is not about the distribution of inputs associated with a given commitment output, but rather the distribution of commitment outputs associated with a given pair of messages.
What you need to show is that for any two fixed messages, the total variation distance between the two distributions of outputs is small. Now, the statistical distance is a metric and so the triangle inequality applies. A good choice for the third distribution is the uniform distribution on $2^k$ outputs. The random oracle model means that the total variation of the distribution of outputs for a fixed message has small total variation distance from uniform and so by the triangle inequality, the distribution of outputs for two fixed, distinct messages has small total variation distance.