I discovered two major details of the Gr”a’s explanation which are missed in most presentations of his opinion. Firstly, the Gr”a does not say that Eli used THE Urim Ve-tumim, but AN Urim ve-Tumim. Based on the Ramban in Tetzaveh, he says that there were Divinely inspired people who used private oracles similar to the Urim ve-Tumim. He says that to avoid the question that I raised about the impropriety of using the actual Urim ve-Tumim to find an answer to a private question.
The second, and most jarring, discovery I made when looking at the Gr”a ‘inside’ is that, ironically, almost everyone who quotes this idea misreads Eli’s misreading! I had always heard that Eli read ‘shikorah’ instead of ‘kesheirah’ – ‘drunk’ instead of ‘worthy’ (here’s an example from Rabbi Frand, here’s one from Ohr Somayach, here’s one from Chabad, and here’s another). The Gr”a actually says that Eli should have read it as ‘Ke-Sarah’ – ‘she is like Sarah’. It’s clear that this is the correct reading. I will translate the contexts in which he uses the word, leaving the word in question untranslated and unvocalized (KSRH): “I am a depressed and barren woman praying for children KSRH”, “You did not give me the benefit of the doubt that it meant ‘KSRH’ who was barren”, “In the Midrash this was explained further – KSRH, Rivka, Rachel”. Is there any doubt?