Dressagey rider here: I have ridden with the halter under the bridle but usually for specific reasons. A young horse who is not yet used to the bit, in which case the halter is used for the transition. Sometimes hang the bit in the mouth with the bridle but put reins on the halter and then slowly work toward using the bit. Lungeing a young horse without a cavesson-- if they are very green or very young, I lunge off of the halter at the beginning as the bit can be a surprise. I like to have the bit be present but not used at the beginning.

As for spooking when transitioning from halter to bridle or vice versa, I do own a dork of a horse who has spooked not once, not twice, but three times during this!!! Twice he broke the reins of the bridle which were looped over his head. The first time it was because I shocked him with static electricity. The following times seemed to be about remembering that I shocked him. Sooo, I never loop the reins over his head while unbridling now. It is very scary to have a horse bolt with a bridle dangling between their legs. VERY dangerous for the horse. So I keep them in my hands and unbridle so that the bridle will come off in case of a spook and all that will be on him is a halter around his lovley neck.