I loaded up an He-111 for takeoff training but when I hit the start button, the Tiger Moth appeared and after that I am unable to even select another aircraft in the training section.

If you are using the "Training" module of game, they are scripted and "work" only in his way...

Use the Quick Mission - Cross Country (allow you take off) and select other aircraft, e.g. He 111.

Previous to this configure the loadout, e.g. for He 111 in Plane and SAVE AS, type e.g. 8xSC250...
In QM main screen select this loadout.

Loadout button on QM - just before fly is only a "check list"...

Quick Mission in CloD is just a "Quick Mission" with only plane selection option - is not a complete Quick Mission Builder.

Sokol1

thank you!