Richard,
Your question leaves many areas undefined in order to produce a complete answer.
For example: the end of your question specifically states spec sheet/msds. So, is your concern for filler material specifications or is it for the msds log and knowing what the hazardous elements and/or conditions are? But, the simple sheet of paper that comes with rolls of wire electrode or in the package of stick electrodes is more than likely NOT what is being asked for.
Now, an answer to your question will also depend upon applicable codes, at least to a degree. The main consideration is if it is for Seismic application or not. In Seismic with Demand Critical welds there are additional requirements that can come into play.
OVERALL, AISC 14th ed, 16.1-118 (J2.6) states "The manufacturer's Certificate of Conformance shall be sufficient evidence of compliance." You can get these off line and they apply to every batch made under a specific product line for the manufacturer. AWS D1.1 is not as clear but does state that if the engineer requests the info the contractor is required to supply certification that the electrode used conforms to the job specs. The main place to always check is in the Contract Documents/Job Specifications. There can be requirements there that will clarify the engineer's desires for the job at hand.
So, in that exact situation, No, it does not make sense. One Certificate of Compliance will cover all the electrode made to that specification. IF you use the same electrode manufacturer and a particular product each time. If you change, you need to go online and get the applicable Cert for the new product.
BUT, if you need qualification for those pesky Seismic Demand Critical welds and are using special steels to boot, you start going down a dark and lonely path that will require further documentation. Sometimes the manufacturer has certs that will be acceptable to the engineer and other times it will require additional testing to prove CVN at the correct ft lbs and temperature, YS, TS, etc.
He Is In Control, Have a Great Day, Brent