Sure it can.
The first number is about low temperature viscosity. The lower -- the more fluid the oil is when the engine is cold (e.g. when you start it in the morning). Less viscosity when cold - better flow and better protection at start. As most of the wear happens when engine is started, that is important.
That said, the difference between 0w20 and 5w20 probably isn't going to matter in most of the places in the U.S, where the night temperature is rarely in the single digits F.
The second number is about high temperature viscosity. Here, lower doesn't always mean better in terms of engine protection - on the contrary. But -- it provides better fuel mileage (almost for free).
All that talk about newer engines having (whatever they now have) -- it's all about newer engines being able to handle lower viscosity oil without breaking down (and it's a cheap way for manufacturers to improve gas mileage). It doesn't mean that 0w20 is necessarily better for the engine than 0w30.
One more thing: if you look at those oils for high-mileage engines, they're all higher weight. Why? Because worn engine needs thicker oil to hold the pressure. Therefore, sticking with 0w20 might be a nice plan, but it probably worth to revise it once you hit some decent tens of thousands. Because it's important for the manufacturer to provide you with good gas mileage and that the engine lasts for the warranty period. After that -- who cares?
Modern engines are good at lasting into low hundreds of thousands, but after that -- it's really your problem, and manufacturer had already moved on, no warranty, the engine is not produced anymore, the car is not produced anymore... Who cares if your engine could've lasted 500K miles but lasted only 120-180K?
But then again, most people don't care for engines which last over 150-200K miles anyway (they're bored with the car after 5-6-8 or so years, the paint is chipped, the interior smells, a lot of things have been replaced and need to be replaced again ...), so it's not worth it.