My understanding is that with iOS, Bluetooth (and possibly WiFi) will be turned on by default when you turn on airplane mode. You can turn off these features after you turn on airplane mode.
But how is that possible? How is that airplane mode in that case?
If the whole idea behind airplane mode is to ensure that all connections to and from the device are disabled, how can you have WIFI and Bluetooth enabled? Maybe iOS just displays them as being still turned on in the system, but in reality they aren't, and all networks and antennas are disabled.
It would be like saying these pair of shoes are waterproof unless you step in a puddle of water. In that case your feet will get wet.
Airplane mode is designed for people who are flying in an airplane (hence the name). There are certain FAA(?) regulations that dictate the types of devices that can be used, and the strength of their signals.
Most (all?) airlines offer very expensive WiFi services for their passengers. Some airlines have also opted to stop installing TVs in the front of each seat and instead offer the same content via their smartphone app that can be accessed via a free version of the WiFi service the airline offers. So you will watch the same movie on your smartphone instead of a TV installed in the headrest of the seat in front of you.
So I am sure that this is not a display issue, and it is the intended behavior. I think "airplane mode" really means that the cellular antenna is turned off.
Another issue is that default settings for some phones allow for someone to turn on (or off) cellular, WiFi, and Bluetooth without unlocking the phone. There may be ways to disallow this, but there is always the risk that someone will find a way around any modified settings.
With regards to the iPhone specifically, Apple is very good with the security of their devices, however, security vulnerabilities are occasionally found. When there are security vulnerabilities found in their devices, Apple is able to patch these vulnerabilities. However someone using an old phone as cold storage would never receive any of these updates, so an attack may potentially use a vulnerability that was found (and patched) a long time ago.