There are significant stabilization benefits to having a public (and binding) inflation target - all of the large economic entities / sectors like banking, government spending, corporations, etc., become more predictable to each other. For this reason, whatever number is chosen will tend to “stick”, regardless of how the actual number is chosen. Or alternatively: it is optimal to have an inflation target (compared to not having one). Once you are in the “targeted” space, there is a further optimization of deciding what the inflation target should specifically be, but the gains and losses here are small compared to the existence or lack thereof of any target at all.
My intuition for this is to view the economy as a large number of interconnected gears driven by an input shaft. When the input is neutral (no inflation), the gears are slack and kind of spin back and forth with occasional whiplash. But with some token input spin (small inflation), the slack vanishes and all gears spin smoothly.
