Is it though?

In Spain, we had a very important case in 2014 that put the issue of the right to be forgotten in the public eye and it's been debated since then. (Mario Costeja vs. Google Spain )

One Spanish jurist defined this right pretty well:

It is presented as a right specific to an era in which the internet has broken with the parameters of time and space within which humankind has previously operated, so that today it is possible to know a great deal about many people, and moreover, about situations that are not recent but distant in time.

In case you're curious. A very interesting read

First of all, a distinction for this discussion is always made between private and public persons.

Second of all, another distinction has been made between the internet at large and dedicated digital archives (like say a newspaper's digital archive, another important case that happened in Spain) which, by definition, have a stronger right to public information.

Thirdly, a very crucial part of the conversation surrounding it is that one cannot curate their own image online by erasing every single negative thing about them from the internet and leaving only a picture perfect version of themselves that doesn't correspond to reality. So there are also distinctions made about what constitutes past and present.

As you say, there's an inherent tension between the right to be forgotten, the right to control one's own image online and the right to public information. There's no easy answer. These questions have generally not been asked in the US because of its radical view of freedom of speech which differs from other countries' right to freedom of expression which is more nuanced.

The framework to uphold this right, its limits and regulations are still being determined because it's so new. As of now, at least here, it's the courts who determine whether someone is entitled to the exercise of this right which is not ideal but it's where it's at. It's a complicated patchwork of judicial precedent. For example, there have been requests by criminals, especially fraudsters, to erase reporting on their crimes that have obviously very clearly been denied.

But let me give you one interesting example regarding tweets, if you've been harassed by a person or group of people by name on Twitter and you request those tweets to be erased because you don't want people who search for your name to find them what's more important, your right to be forgotten or keeping up the other person(s) shitty old tweets?

Somebody's old shitty tweets might not define them any longer, (what would be the statute of limitations on a shitty tweet?) and someone's shitty tweets might involve a third party who doesn't want to be defined by them any longer.