https://www.nationalreview.com/2017/04/al-gore-lincoln-quote-fake/
It existed at least since 1908, so if it's fake, it's a very old fake, and written at a time when many of the people who knew Lincoln were still alive.
It would seem to me that claim that it is from a letter to Elkins ought to be verifiable.
Also, I cannot see anything at your link. My old browser on my old computer has stopped working at many websites, and apparently National Review is one of them.
If you could copy and paste it in this thread, I could read it then.