Of course, the real problem is that the site administrator is allowing Twitter to run arbitrary code on their website. I have much to say on that topic, alas, it is the choir reading HN and they don't need me to preach.
That sure seems hostile.
If I were still actively using Twitter and posting something that I later delete - OK. I can see the argument from a 'right to be forgotten' point of view. If the president of the US does so to beautify the public record, not so much.
In Germany we have the concept of a person of public interest. People who are publicly visible due to their position in politics, media, entertainment or because they try to have their face seen by any camera lens available, need to "endure" more public/media scrutiny than Jane Doe from next door.
The problem is, that this can only be decided (in a conflict) by due process in a court of law.
So Twitter deciding to use privacy as an argument to enable a few reported on whales to beautify their record (because let's be fair, who embedded a tweet from Jane Doe?) seems at least questionable.
It's not just a few whales. Imagine you're being arrested, someone films that arrest while the cops are speaking your PII (name, address) into their radios, and the video of the arrest ends up on Twitter.
Under "right to be forgotten", you could (and can) demand removal of that video - but your data would still be floating around the net.
Do I have to post the quote?
"your rights end when they infringe on the rights of others"