Eccodecco - be careful removing the throat latch, it is there for a very good reason aka to stop the bridle falling straight off if they scratch their face or such. I was trail riding with a friend a few years ago, both on our ottbs. She was not overly into cleaning her tack and as a result her bridle was quite stiff and cracked.
A kangaroo jumped out in front of us and her horse spooked quite violently. The throat latch snapped, and he took off flat out up the track. As soon as my friend tried to pull him up, the action on the bit pulled the head piece over his ears as he ducked his head down, and the bridle came off.
Hence, I would NEVER ride if my throat latch was looking a little 'unhealthy', let alone without one altogether.
As for bits, my horse is currently in a loose ring french link snaffle, but is very fussy in the mouth so I have been recommended to try a hanging cheek/baucher which apparently will stablise the bit in his mouth a little more.