Why does density equal mass divided by volume?

You could define a quantity (call it rareness say) = volume/mass. So density = 1/rareness and rareness = 1/density. But in common speech, we (even if we don't know any physics) say lead is denser than feathers.

That's simply the English language meaning of 'dense'. So if we measure density we expect the value for lead to be a bigger number than the value for feathers - not a smaller number. So density is defined as mass/volume so it matches the common English usage.

The same argument applies to speed. If A is faster than B, the calculated value of A's speed should intuitively be a bigger number than the calculated value of B's speed.