If you are always changing oil at 3K, IMO synthetic won't make much difference. As I understand it, traditional multi-viscosity oil has many additives to make it have the properties it should. As oil is heated and pressured in use, the additives start breaking down and the long chain hydrocarbons start gradually shearing into shorter chain molecules. The oil's ability to protect gradually decreases over time, plus the detergents get compromised and their ability to deal with contaminants declines. But all this takes time (well, use). By 3,000 miles the deterioration hasn't gotten that significant yet. If you waited till 5K, then yes you might see a benefit in less wear.
One thing though, is the synthetic oils will keep seals in better shape. With ordinary oil they tend to shrink and get somewhat less pliable. So synthetic could help there. You might not start using oil as quickly.
A good synthetic like Mobil 1 or Amsoil (I've used both) start with a premise: what characteristics do we want this lube to have? And they manufactured molecules that will do those things right from the start. Whereas a conventional oil starts out as crude, gets refined, gets poked and prodded and added to, all in an attempt to get it where it needs to be. Some of that prodding and adding starts to get counteracted the first day it gets circulated in the engine. Synthetic is sort of a more elegant solution to the lubricant problem. But we pay a correspondingly higher price.
I think the biggest thing one can gain from the good synthetics is extended drain intervals. 15K, 25K miles, maybe more with enhanced filtration systems. If a person isn't going to run a synthetic longer, its cost can be quite a bit higher.