Trying to figure out how to refer to Solzhenitsyn, I did a Google Ngrams search comparing Alexander Solzhenitsyn, Aleksandr Solzhenitsyn, and Alexandr Solzhenitsyn. Alexander (the translation) was in the lead, though the transliteration Aleksandr has recently been almost nearly as popular. Alexandr is very rare.
For 19th and early 20th century Russians that have easily translated names, Leo Tolstoy wins hands down over Lev, Peter Tchaikovsky handily beats Piotr, and Czar Nicholas beats Czar Nikolai — yet Nikolai Gogol is vastly more common than Nicholas Gogol.
In the 20th century, Leon Trotsky is much more common than Lev Trotsky. Joseph Stalin is also much more common than Josef Stalin, though Josef Stalin forms a substantial minority (about 20%). Yet Josef Stalin is neither translation nor transliteration; the Russian pronunciation would be roughly “Yosif” or perhaps “Iosif,” which are extremely uncommon. But Mikhail Gorbachev is the standard, with Michael being almost never used.
In any case, I just thought I’d note this, in case people have some theories. I suspect that there’s something of a longterm trend towards transliteration rather than translation, but that doesn’t explain everything. There is also likely some effect based on the Latin-alphabet language in which the people first became known (perhaps that’s why Tchaikovsky got his initial “T,” unnecessary English but necessary in French). But I don’t think these explain everything, and I’d love to hear other explanations that people might have.