Why does "airplane mode" turn off Bluetooth and WLAN if I can just turn them on again? On my (Android) phone, enabling airplane mode automatically disables all wireless antennas except for GPS and NFC (I assume the latter is either passive or way too low power for anyone to care about).  In particular, it disables mobile data (4G/LTE, or whatever it has roamed onto), WLAN (WiFi), and Bluetooth.

However, I can turn WLAN and Bluetooth back on again without disabling airplane mode.  Asking the airline, as suggested in this answer, is useless.  During the safety briefing of every flight I remember in the recent past, they would say something along the lines of "You can use electronic devices as long as they're in airplane mode," or very rarely, "You need to switch your electronic devices completely off until we hit 10k feet, and then use them in airplane mode."  But they don't tell you anything more specific in the safety briefing or on the information card (e.g. "WLAN is fine, but keep Bluetooth off" or similar).  At the same time, most airlines clearly want you to turn on WLAN so that they can sell you internet access.

The obvious interpretation is that you can do whatever you want as long as your device is in airplane mode ("They said to put everything in airplane mode, and I did, so I complied with their instructions.").  But that actually doesn't make sense.  I own multiple devices which only have Bluetooth and/or WLAN antennas (e.g. the Nintendo Switch, most smart watches, and probably a truckload of other non-phone devices).  In most cases, you can turn both antennas back on after enabling airplane mode.  At that point, airplane mode is just a pretty little icon that appears in your device's status bar.  It doesn't actually do anything (but is presumably still required).

What is the point of disabling Bluetooth and WLAN if you're not prohibited from turning them on again?  What is the point of even having airplane mode on devices that don't support mobile data in the first place?
"Airplane mode" is a convenience feature provided by the phone manufacturer. You can choose to turn it off anytime in violation of the rules, so there is no difference there. 

What it does may not match the regulation in any particular geography as rules vary. Regardless of any particular device the airline sets rules to ensure that no device: 

1) Interferes with navigation signals required by the aircraft, particularly at landing

2) Confuses cellphone towers on the ground which expect radio signals in roughly a planar orientation near the ground

The airline gets its rules from the FAA/EASA... and indirectly from the FCC... What was considered "airplane mode" by the FAA changed in 2013. Testing determined that wifi and bluetooth did not cause sufficient interference with ground based nav signals, so that prohibition was relaxed. An old phone's "airplane mode" may be too restrictive.

There was no change to the rule on cellphone voice communication, which remained prohibited. Once the aircraft is sufficiently high no ground based cell tower can receive any signal, so this prohibition mid-flight is effectively meaningless.