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.
I've been able to archive them for a while now. The problem is it's only a single post and I often want it and the comments that go with it.
I've found that Ghost Archive is better for preserving long Twitter threads post-Musk.
I will test that out. I've been using Thread Reader and random nitters.