I don't think most of the objections are normative (ie, free speech and firearms are good) so much as empirical (ie, the nature of the internet, or of human networking in some more abstract sense, precludes a "right to be forgotten").
It's a peculiarity of the current age that we can find out so much about people so easily, that trivialities are preserved in photographic quality, and that these records are preserved publicly for extended periods of time.
I think the "right to be forgotten" has always implicitly existed and that we accidentally erased it before we realized what we were doing, and only now are seeing a backlash as the effects of that choice are being felt.
What's worse is that back then people had difficulty discovering rumors about themselves. At least today, we can do vanity searches.
No one is saying that people don't have a right to their memories. Even communities have the right to remember things collectively. But imagine if you couldn't get a job in your small town because everyone whispered about you being a child molester or something. You couldn't get housing. No one would talk to you. You had to beg for food. My example is a bit extreme but it illustrates the example well because some of these things happen on a massive scale to people whose potential employers web search them.