Yes bro. you are correct. turning off BT or wifi is temporary when we do it via control center. If the bluetooth is off permanently from iPhone settings - bluetooth, then you may be able to use the watch in cellular mode. but to save battery watch connecting to iphone using the data will be good choice too.
but we have to manually go inside cellular toggle and check if its connected or not. try making a phone call to confirm the same.
through which app you are testing internet connectivity bro? apple music?