For anyone else with 360 controller mapping (particularly left trigger) it seems there are a few with the issue, there's also a fix
Finally I can fly jets. Looks like freelook in the chopper is still borked though.
http://answers.ea.com/t5/Battlefield-4/BF4-PC-Duplicate-key-bindings/td-p/1729315/page/2
1. Make sure BF is not running.
2. Open PROFSAVE_profile file in a text editor. The file is in Documents\Battlefield 4\settings folder.
3. Locate the following entries in the file:
GstKeyBinding.heli.ConceptZoom.3.button
GstKeyBinding.jet.ConceptZoom.3.button
GstKeyBinding.vehicle.ConceptZoom.3.button
4. Change the numeric values after each of these entries to 255
5. Save the PROFSAVE_profile file.
6. Enjoy!
NOTE: if on starting the game first time after the change, Origin reports that your local saves do not match the saves from the cloud, choose to use your local saves (no those from the cloud).
NOTE 2: for my xbox controller I had to set all GstKeyBinding.heli.ConceptZoom from 3 to 16 to the value 255, to make it work for heli. Although, it completely disables free look for helis.
Thanks to all who figured this out!