Even though we are not currently traveling, my spouse and I have discussed what would be done with phones when do we (eventually) resume travel (or even resume getting out among dense crowds here at home, which sadly has become a place of pickpockets). My husband always puts his phone in his front shirt pocket, which in my eyes is not the safest place, especially for likely pickpocket situations.
For traveling, I think it would be best placed in the back (open) pocket of the ole Rick Steves' neck wallet, then worn under his shirt. Somewhat easy to pull out of the open pocket without having to unzip, and somewhat easy to pull up from his buttoned shirt, maybe just unbuttoning one or two buttons. Of course, I am SURE my spouse would consider that a pain when the time actually comes......but what is the bigger pain? losing the cell phone in another country or dealing with the neck wallet storage?
But, that said, in reality, when traveling, I know he would want his phone in his hand, ready to take photos at any point in time (other than when on public transport or moving thru dense crowds), as likely your sons would be.
One person above suggested having the phones left in the hotel safe, but if your group gets unexpectedly separated, the ability to communicate via text would be easier vs. trying to find the "lost" person(s). And, access to translation devices, maps, etc. on the phone would be comforting.
Is there such a thing as a cord device that can be securely attached to a phone to allow easier access than a pocket in a neck wallet? If so, I am all ears....................