Ahh cool, I have an older/diff model without the BT bit but luckily it has a battery indicator on the face.
Yes, the Bluetooth on this is a little bit gimmicky really. It can be useful to set the watch up and can be used for time syncing from your phone, as you say, but there is multi band 6 for that anyway. It can also be used to find your phone and for remotely setting way points, using your phone's GPS (E. G. If it is on your rucksack). The only time I use it, is very occasionally to check the battery level in the winter (as its not actually indicated on the watch).
The Bluetooth versions allow you to change the date order.
BTW, manual is here for the module I believe: https://support.casio.com/global/en/wat/manual/3461_en/KPNDSYybldrnhw.html Found it from https://www.casio-europe.com/euro/products/watches/g-shock/gw-b5600mg-1er/
Actually, the opposite. It syncs time from your phone to avoid drift it seems, that and can alert you of specific dates/reminders etc. Over a £4.99 watch, it's a lot better made and will last (based on experience), will charge from the sun and will avoid time drift either via bluetooth or time signal here in the UK and a few other locations around the world. Design is polarising and won't be for some, but I love these style G-Shocks... all the info you need (time & date) in a very tough shell. Banged mine of all kinds of stuff over the years and it still looks like new. Not sure the bluetooth adds much though.