Everyone already has the right to remember something. It is as simple as saving the web page or printing it to PDF.
How can anyone exercise their "right" to "be forgotten" if saving local copies is allowed?
Nobody can stop you saving it.
Although this particular legislation doesn't ban that, laws already exist making it illegal to make local copies of certain content. Plugging the "local copy" loophole would be the next step.
The summary says that LINKS to outdated and irrelevant information should be removed on request. It doesn't say anything about the data itself.
You left out the quotes around the word "irrelevant", which were there because it's subjective. Who gets to decide that? Same for "outdated", even there were no quotes in the article.
Thus if a newspaper publishes a story about you being drunk at college when you're 21, in 20 years time you might ask Google to delete the link from its cache (the link is now to outdated and irrelevant information)
Even if said person is about to run for public office? (Just one example.)
but you can't ask the newspaper to withdraw publishing of the article for it owns the copyright, etc. Now it might be hard to find that information once the link is removed by Google but that's another matter.
If the newspaper is allowed to continue publishing the article, then the incident isn't "forgotten". Another loophole. To "protect" people's "right" to be forgotten, it's necessary to ban search engine links to the content, posting the content itself, and the making of local copies of it. If any copies survive, anywhere, the job isn't done. On the other hand, if your belief is that the goal should be to make access not impossible, but merely difficult, that just means that only the rich and/or powerful will be able to find the information. How is that a good thing?