What about using a tablet or suchlike device- at least you'd be spared the rustle of papers while other folk are singing.... don't see how you'll get people to stop faffing about at the crucial moment
for whatever reason.
In another context, how many times have some of us stood at a supermarket checkout while the person in front, having done everything else, suddenly realises that he/she also has to pay, and that their purse/wallet is buried deep in their bag or pocket? It's just human nature to faff, so get used to it!