Just get good stock from the beginning. If you start with healthy, calm mice, you will continue to produce healthy calm mice. Feed off any that are not handle-able or that bite, and you will keep your line nice and clean.
Babies from tame parents don't need to be played with daily to raise up tame. Mice are highly intelligent animals, who have the capacity to enjoy being handled and enjoy enrichment activities though. So don't avoid contact with them just because you can. I would definitely recommend being as comfortable as you can with handling them, because you never know when you may need to handle them. And if you pre-kill your food (which I'm of the opinion everyone should, but that's another can of worms) then it's important to be comfortable handling your mice, and knowing how they act, especially if you do cervical dislocation, and not CO2.
Also about preventing the spread of diseases? What are you talking about?
Pet and feeder mice shouldn't be infected with any sort of diseases.