I very rarely use saddle soap.
Bridles are fully rinsed under a running tap after every ride, where I quickly rub off gunk and sweat. Then they are just shaken off and hung up.
I only really deep-clean them about twice a year. Whole thing is taken apart and soaked in hot water for a while, then I scrub them with a nail-brush and whatever saddle soap I have on hand, rinse and dry with a towel, hang up and leave to dry for a while. Then I condition them with my favourite leather conditioner of the moment.
I’ve got ten-year-old bridles that still look like new, so it must be working
Disclaimer - Climate where I am is dry, no ice/snow. Don’t know if that makes a difference?