We rotate the deal and always run two decks, but we don't have a rule about who shuffles...
Someone who isn't currently dealing shuffles. Whoever won the last pot doesn't shuffle. Whoever is really bad at shuffling or hates shuffling doesn't shuffle. Someone else always does. It always works out. Very often, the person who just finished dealing shuffles, because they had the stub and the burns nearby, and were the natural person to scoop the rest of the cards... but if the dealer wins the pot, someone else will grab the cards and start shuffling them.
The only hard and fast rule we have is that someone must cut the cards other than the person who deals them - and, critically, only a single over-hand cut onto the cover card is allowed. (Virtually impossible to do a false cut that way, so even if the shuffle was controlled, the top is lost.)
We use the cover card to keep track of whether the deck is ready to run... after shuffling, you put the cover card on top of the deck; that's the sign that it's ready to hand to the next dealer. Dealer peels off the cover to whoever he offers the cut to; that person cuts onto the cover cards and the deal begins.
If a dealer is handed a deck without a cover card on it, it means it wasn't shuffled...
We also don't make everyone deal - some people aren't comfortable. And if someone just won a hand and is stacking chips, but is due to be the dealer, someone else always jumps in and deals for them. My crews includes a lot of long-time card players, and keeping things moving seems to come naturally.