After Twitter migrated to the retarded x domain name (if you had an account, you were forcibly moved in the past week), it is now once again possible to archive tweets with Archive.today - archive.org still fails, unfortunately.
Obviously, you have to select the x domain name though.
Might have been a technical issue. I use an old browser on one of my PCs and embedded tweets on this site haven't worked for months, but they suddenly started working again today.