Yes, SS's diagram will power on when turned "off" to charge IF you plug in a wall power adapter to the wall power jack.
SS's diagram utilizes 2 power input jacks. One for battery charging and one for wall power adapters. So, the unit will NOT turn on if it is switched "off" with just the charger plugged in.
If you still want to do like you say using DPDT for the first and second switches is totally redundant. Just use SPDT's.