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The other important thing is to reduce uncertainty as much as possible. You don't want your inflation rates to bounce around like crazy from day to day. Instead, you'd like to be able to keep them within certain limits so people can plan ahead to the future.

If I'm not sure what inflation will be day-to-day, I can't make an accurate prediction as to whether I should save my money or spend it.

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And if inflation is really high and fluctuating everyone makes the decision to never save in cash which isn't healthy.

But a bit of low level inflation is generally seen as a good thing. The questions over the efficacy of monetary policy as a stimulus tool in all situations is certainly questionable though.

It's also worth noting that independent central banks are a fairly new thing. Inflation levels used to be set by politicians who would notoriously let short term interest over ride price stability.

The Bundesbank was the first independent bank drawing lessons from the horrors of hyper-inflation. Most (all?) developed countries now follow that model.

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