Everything everyone does is part of history.
Actually, that's not at all true, at least in the meaning of "history" before the internet. History is traditionally a narrative created about the past, usually derived from reliable sources (or at least what were considered reliable by the author of the narrative). A random recollection of some dude about some other dude was not "history" -- it was "gossip" at best. It only became "history" if someone wrote down the account and gave it credibility.
In the past, reliable records about the vast majority of events and people were scant. There are major figures of medieval Europe, for example, where we have almost no actual records from their lifetime -- maybe a baptismal record, or a record that they were paid once by some guy at some point, but that might be it.
The fact that little Jimmy went pee in his pants during gym class in 3rd grade didn't used to be "history." Maybe a few of the kids in his class might remember that incident a couple decades later, but it was generally forgotten by everyone else. Nowadays, one of those kids might take out a cell phone and take a picture of little Jimmy's wet pants, text it to some other kids, and the picture might end up on the internet if it's sufficiently entertaining to some stupid kid.
Now Jimmy's pee-filled pants are an official durable record that might persist on the internet for decades, available to anyone with sufficient skills at searching.
We used to have a historical "filter" that would get rid of the random quotidian minutiae of our lives, simply because it wasn't recorded in durable form. "History" would only record "important" stuff.
Now just about any event can be photographed, videotaped, or otherwise documented to become a "meme" or at least passed around among hoards of people (and thereby become a somewhat permanent record).
The problem here is that we ALL do crap in everyday life that would look bad out of context. And once that crap "bubbles up" somewhere on the internet, it really does become a part of "history" in the old sense, because search engines are our new machines that curate historical records... rather than historians digging in archives and collecting records which would be turned into a narrative.
I'm NOT saying any of this is "bad," only that is VERY different from what "history" was even just a couple decades ago.
It starts with misunderstandings and people saying "they were a kid when they did that" and ends with inconvenient facts about what people did before their "views evolved" being forcibly erased for the convenience of the one wanting their past hidden.
You have a good point, though I doubt that anyone can succeed these days in having something "forcibly erased" from the entire internet AND all public databases AND all paper records.
What some people are proposing -- and what people are asking for in the "right to be forgotten" -- is to consider that some information be removed from prominent locations in major search engines, which (as I said) have become our default curators of "history." Note that it is "curating," not merely keeping records -- search engines need to decide what the top links are. And the algorithms they use may bring undue weight to random events that would largely have been forgotten a couple decades ago.
To be clear: I think the "right to be forgotten" actions against Google are NOT a good solution to this problem. I don't have a better solution myself either. But we do need to recognize that we live in a different world, where "history" is very different than it was just a few years ago. How we deal with that is yet to be determined, but our social mores and standards certainly haven't caught up yet in how to evaluate the new kind of "history" available to us.
And making some rant and slippery slope argument that making search engine hits less prominent will necessarily lead to the "forcible erasure" of history is just ridiculous, especially in the age where anyone can duplicate and store information in multitudes of places on the internet.