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It's not that we don't use mode, there are definitely times mode is used. It's just that mean (and median as well) contain a lot more useful information about distributions that we often care about. For a normal distribution mean, median, and mode should all be identical. So why do we use mean? Because mathematically, the mean is what underpins the formula for the normal distribution, not median or mode, and when you're talking about doing math with normal distributions mean is the thing to talk about (along with standard deviation).
We use median a lot too, you probably just don't hear it called median very often. The median is useful in non-normal distributions, and it defines the 50th percentile, so along with the 25%-ile and 75%-ile you've got your quartile distributions. We use these all the time to talk about grades in schools, or when we talk about home prices distributions in a given area, or salaries within a given field.
We use mode too, again just by a different name most of the time. Any time you've asked "what's the most common blank" you're basically asking for a mode. When we talk about "average" income in a country, we're usually actually talking about median or mode. Favorite animal? Answered as a mode.
You have to use the right statistical tool for your question: unfortunately English doesn't do a good job of conveying this without math jargon.