Because they mean very different things. Imagine you tallied the spending of 5 people in your restaurant:

10 15 30 100 150

First of all that distribution has no mode, so let's then check the next 2 customers.

10 15 20 30 100 150 150

Cool, now checked with 7 people this, and we can say the following.

The mode is to spend 150. Almost no one does this, but that is the mode regardless.

The median is 30, this tells you that half the people spend more than this, and half the people spend less than it. However it doesn't give you an accurate idea, because the people who spend less spend close to it, but the people who spend more spend way more. So if a guy spends 35 he would look like a high spender, but in fact he probably should be in the low spending category.

The average is 67.85, no one spent this amount, but this tells you that if a person spends more than that he's a high spender, so of someone came in and spent 35 you would know he's not one of your high spending customers.

Now let's see how each of those numbers is at predicting how much 7 customers would spend, let's look at the same values, where the customers spent 475. The mode tells you that people will spend 1050, that's absolutely wrong. The median tells you that they'll spend 210, that's also very wrong. The average however tells you that they'll spend 475 which is the exact number.

This is the same for every other statistics, even if it doesn't make any sense to say that people have an average of 2.3 kids, if you were planning on receiving 10 random families they would probably have 23 kids in total. Average is good at predicting large groups, and that's the information we usually care about when we're trying to express a large group in a single number. If you want a second number the obvious choice is the standard deviation, in the example above the standard deviation is 63.76 this gives you an idea on how accurate is your average at predicting, so in the case above not very accurate at all, but if we imagine that the number of kids above had a standard deviation of 0.2 you can be 68% certain that the 10 families will have between 21-25 kids, or 95% certain that they will have between 19-27 kids, or 99.7% certain that they will have between 17-29 kids. Working with the level of confidence in a prediction allows you to evaluate certainty at doing things. If you only knew that the median was 2 kids or that the mode was 1 kid you couldn't predict things with any accuracy.