The "right to be forgotten" is the "right" to suppress true speech by others, and is in fundamental opposition to the right of free speech. Its a close cousin of the forms of libel laws where truth is not a defense that suppress true and accurate criticism, and the recognition that you can't have that kind of thing and free speech at the same time isn't a 21st century observation.
Its also, from a pragmatic standpoint, about equivalent to attempting to command the tides by law.
There's a distinctly 21st century flavor to being able to store all the banal life mistakes indefinitely, to dig them up later if we ever need dirt on a particular person.
Before it was varying levels of unprovable (ie, not photographs or verifiable sources) and bandwidth limited (people could only store so much information; sources weren't interlinked for quick queries).
So it's actually quite likely that when we drafted our laws, this was an implicit right by the nature of technology, and considered a natural part of the bounds of specifically enumerated rights.
We should make sure, as technology removes traditional limits on what we can do, that we adequately adjust our laws to reflect that change.
Except not really. Google, despite what many might like to think, is not an "official" entity, it's a private company. They aren't disseminating information for the public good, whatever that is, but so they can slap an ad on it and make some profits. So what if Jo(e) Bloggs did something embarrassing a decade ago? If one of your friends was constantly bringing up things from years ago, you'd get tired of them doing so pretty quickly. Why should Google get to do it to make some money off of it? And what sort of gossip monger even wants this to be possible?
So the New York Times isn't a private company then? Just repeating "private company" is meaningless as a justification for well...just about anything. You'll have to find a different point to argue.
The NYT did the work and wrote the story and the editor decided it was worth publishing. The don't just dredge it up, slap an ad on it, and call that adding value. And if a newspaper gets something wrong, it does retract it.
Its also, from a pragmatic standpoint, about equivalent to attempting to command the tides by law.