We have learned that it is possible to request an earnings history using Form SSA-7050. However, it could cost up to $80 if you want an entire 40-year earnings history. The form notes that it is primarily used for non-Social Security purposes such as pension plans or personal injury suits and that if you want non-certified totals of earnings, just request a free statement using Form SSA-7004. But then, of course, if you search for Form SSA-7004 you are told that the Social Security Administration is no longer honoring statement requests.
It's probably not worth it to request earnings unless you have reason to believe specific years' earnings were reported wrong. I have found that errors are rare. When they do happen, it's usually when earnings are reported by tax return for self-employed individuals rather than by W-2 for employees. To be safe, I would advise all who still have their last statement to check the earnings records; if they think one or more year's earnings may be wrong, they can then request a certified copy using Form SSA-7050 and pay the $15 or $20 for just the years in question. Or they can wait until SSA starts posting statement online, which is supposed to happen sometime this year.