Why don't you call your broker and see what they say to this proposal?
Honestly, I'd like to know.
Your buys and sells are legal documents as well...as everything is supposed to be reported to SEC and other government agencies, so I don't see how a piece of paper attesting to a certificate would be any better than the current statements of your brokerage.
As for holding certificates meaning you are out of the action, this is true but if delivered back to a broker they should be tradeable in 1 to 3 days.