Legal name and social media name don't necessarily have to match.
I'm in the process of transitioning to a single username everywhere based on the domain ngp.io, which itself is essentially "[my intials] I/O". I like the elegance and conciseness of it. But I'm extremely glad that I didn't do something like this earlier. My childhood online (as well as my "online childhood", if you catch my meaning) remain obscure to all but those with the necessary time, resources and motivation.
That's surprisingly powerful when you're surrounded by swarms of people whose entire histories can be looked up in a matter of minutes.
It can also be damning; I'm reminded of the line in Mad Men referring to Don Draper, "He has no people! You can't trust a person like that." Being the mysterious man from out of town might be alluring from a sexual stance, but business-wise it's still difficult to advance without strong geographically-rooted social networks, and the first step to creating those nowadays is often glancing at someone's Facebook or Twitter and exclaiming "hey, I know that person too!" Personhood is partially defined by your connection to other people, so having a visible interpersonal history is, in a sense, required for passing the Turing tests you're subjected to by those around you. Gaps in your history, on purpose or by accident, can result in being treated like a subhuman.
There's always been a feature of childhood that I feel is paramount: the freedom to alter one's identity. The child is still in beta; they're a person that's not officially released to the public.
Forcing them, from the start, to stick a single identity can be problematic psychologically and socially.
Encouraging situational pseudonymity is not equal to the revocation of one's legal identity, from now until death.
I can say with certainty that, had I not had the benefits of pseudonymity offered by the internet of the 90s and 00s, I would be a fundamentally broken person.
Then we should work to make that future unlikely. Even if things come to be as you describe, we should build in strict social/technical safeguards of (at the very least) pseudonymity until we much better understand the development the human mind.
[edit in response to your edited post] It seems we're pretty much in agreement, just coming at the issue from different points of devil's advocacy.
I agree. I'm very grateful to have a common name. Nate Perkins is about as generic as it gets. Global searches for my name come up with a variety of people completely unrelated to me.
Still, there are potential downsides. For instance, searching for "[my name] [my city]" brings up a mugshot of someone who isn't me. And of course, Stephen Law of the original post has a much more frustrating issue.
None of us are really in a position to say with certainty "it's better to have a common name than an uncommon one" or vice versa.
I would argue that pseudonymity is necessary to overcome negative situations arising from having a common name or having an uncommon name. But this argument is strictly from my own experience and should be taken as such.
Like most things, it comes down to the importance of teaching kids to think critically about how their choices will affect themselves and their environment rather than accepting truisms not backed by relevant statistics.