Airplane Mode does disable cellular, WiFi and Bluetooth; however, you can "lock" your device onto a specific network in Airplane Mode to send/receive data, such as to only send/receive on an in-flight WiFi network or hotspot. Airplane Mode also disables roaming.
There may be a lot of culprits causing you to not connect, but it'd be easier to not use Airplane Mode.
From what I've read about Airplane Mode is once you've enable it, you'll be disconnected from everything wireless. After in Airplane Mode, you'll need to connect to a WiFi.
What is the purpose of Airplane mode in general? I know it is to avoid interference with the plane, but that seems unlikly. Wouldn't the cell towers at the airport interfere with the plane more then a single cell phone? Kind of seems like the no cell phone at gas station rule. Why no "Gas station Mode"?
Airplane Mode keeps evolving. It's original intent was to terminate all radio processes on your device; however, you could still use the device for apps not requiring a network connection, such as games, pictures, spreadsheets.
Then the airlines started offering in-flight WiFi and I've read in-flight cellular soon.
If there is one place on earth with the most radio signals, it's probably an airport. You're right, cell phones would be the least of any interference because airports have signals emanating from other aircraft, ground crews with handhelds, cell phones in the terminal, the tower and all the Air Traffic Control and Landing Systems (ATCALS) equipment you'd see at the end of the runway is also radio signals. Once in the air, however, it probably clears.