LeBron's strength isn't really a factor on the perimeter. He gets past people for two reasons (which apply to everyone):
1. Attempting to cut off a ballhandler in one direction (i.e., bumping him on an attempted drive) is dangerous because he can easily change direction, and now he has an angle on you and you're out of position, and
2. It's not worth it to attempt to body up a ballhandler and possibly be called for a foul
Even the best defenders usually don't try to stay perfectly in front of a ballhandler-- that's how you get crossed. They just try to make the ballhandler take the angle that the defender wants them to take.
EDIT
Here's a great example:
Tim Duncan is perfectly squared up with CP3 when Paul gets past the first defender. Duncan may be old, but he's still a professional athlete and has a distinct size advantage on Paul. If Duncan wanted to, I bet you he could have made an attempt to slide his feet and stay in front of Paul. Instead, he turns his hips and simply allows Paul to take an angle that still allows Duncan to contest.
Obviously, as you can see, using this defensive strategy properly doesn't always result in success, but it's still there and it's what NBA players do. They don't try to bully players on the perimeter, even when there's an opportunity to do so.