Not at all. Social standing is up to each individual person based on their own quality and behavior. The important thing is that there is a free system in place when they are born.
Voluntary taxation would be the bedrock of such a system.
Here's the problem, say you have two people, Joe and Ariich. Both are on equal footing socially. They each have say 100 dollars, no land and no assets. Well by the end of their lives, through hard work and good decisions, Joe now has 100,000 dollars, and a good house. While Ariich has 10 dollars and a shack.
Now, they each have a child, Joes child is born with 100,000 dollars and a good house, while Ariichs child has 10 dollars and a shack. Now, Joe takes that 100,000 dollars and buys land, or whatever and by the time he dies is worth 10,000,000 dollars and a mansion and lots of land. Ariichs child couldn't have done that with what he had, so he ended up with 100 dollars and a mediocre house due to hard work.
Joes child has a child, who now has 10,000,000 dollars, a mansion and lots of land, while ariichs grandchild has 100 dollars and a mediocre house. Now, do they both have equal oppurtunity? Of course not. And they didn't earn any of it, they were simply born into their circumstances.
This is one of the problems with your fantasy land.