I don't use the Social Security Death Index. In fact, I rarely use internet sources inasmuch as most websites are woefully incomplete and/or inaccurate. The much-touted SSDI, for example. Several sites offer the SSDI (at least three companies offer it for sale on CD ROM). What you are NOT told, however, is that what is offered is merely a random sampling from certain areas after acertain date; almost all data previous to that date must be ordered from the Social Security Administration (at $7.50 to $16.50 per inquiry - which is the main reason SSA consistently refuses to release complete records).
I've been doing this for over 40 years - much longer than personal computers have been available. My main resource is a network of contacts I've developed over the years. One contact, a retired state employee, lives in Ohio near the West Virginia line. I just received a message confirming the error you mentioned. She also informed she is going "over the line" tomorrow and offered to run by and PERSONALLY check records and, if on file, obtain a copy of your grandfather's birth certificate. My cost: nothing. Your cost would have been the same.
I borrowed this from Rudyard Kipling: "Very often we do or say something stupid (my rendition) when that which we seek is just within our grasp." I suggest you sit tight and nurture that empty purse - and this goes for all you others who are equally "poor" because you're poverty extends well beyond your bank account and is greater than you'll probably ever realize.