Unless you have really sandy soils. Then the rain mostly passes right through (ignoring all the parts that are paved - insta-flood!), and it's just a situation with malnourished grass, probably more susceptible to "weeds", and not a lot of organic matter to hold all that soil together.

That's the thing missing from this image - it's not just being more porous that makes larger plants retain water better, it's that they're a critical part in creating the conditions that produce more of that organic matter, getting that carbon (and a lot of other stuff) in the ground. It acts as a sponge, and in sandy soils that are too porous, it fills those gaps and acts as a binder.