Thanks everyone! I have the right idea with the process, but I think I'll replace the oil with one of the mentioned waxy/conditioner products. I find the oil makes it look/feel nice very shortly, until it kind of dries in/soaks in. Then the next day it seems stiff again.

My process is a wipe down with damp sponge, take the bridle apart (seriously easy since I don't use a noseband ), wipe down the bit with the sponge before getting it messy with cleaners. Then I typically would go straight to the oil unless the bridle was somehow really dirty, then Lexol cleaner first. Oil a few times with time to soak in between applications.

My bridle is stored in a nylon bridle bag of sorts, that hangs on my horse's stall. The floor in the barn is dirt and REALLY dusty in dry months. So that keeps the dust/dirt off my leather tack (bridles and halter). But it's still a rather dry environment mostly.

Phew, still glad I don't have to go through this process with the saddle though. I used to be horribly anal about cleaning my saddle all the time and it took forever. But I was a teen with nothing else to do. Plus, the need to ride in the rain and snow scares me out of a leather saddle.