I've always been told never to oil strap leather goods. They can stretch and deform.
I bought a very expensive, hand tooled bridle made of very high quality leather. It was soaked in oil overnight (not by me!) and for YEARS leaked oil on me and my horse. I have the ruined gloves to prove it!
It has now finally started to act like leather instead of a soaked sponge after more than 6 years.
I have a double bridle from the same maker, in the same leather and all that has been used on it is Passier lederbalsam and Fiebings Saddle soap. Because it is high quality leather, it was easy to soften. It has not stretched (like the other bridle) and one can actually grip the reins!!
I highly suggest using a high quality leather conditioner. Leather CPR also works very well for poorer quality leather.