That is a good point, Richard. The same can be said of the U.S. Social Security Death Index, and pretty much any other record whether they are considered primary sources or secondary sources. That's why it's important to document every source found and to look at the big picture when researching.
Birth and Death Records in the U.S. have a shaky beginning, and the culture of reporting information to the government didn't exactly match what we experience today. Modern records are much more accurate, although I think a huge part of the reason for this accuracy is simple: Most people are born and die in hospitals now. Still, the accuracy of public records rely on the recording clerk to be diligent and reporting relative to be both honest and knowledgeable.
That's the actual record or certificate. As you've pointed out, the chance for error increases dramatically when those original records are then indexed. County Index books for marriages, births, and deaths can be really helpful - but it must be remembered that this information is being transcribed from other (hopefully original) records by a government employee who is human and therefore fallible.
And then... the internet arrived. Did it make searching public records easier? Yes, without question. It's fantastic. Of course now we have even MORE indexes, and these are usually done by volunteers who are, again... human.
Indexes are great.. but better to look at the original document when you can, and measure your confidence in the information you find against what you've found elsewhere.