I would not recommend it on an OEM strap, but I make my own straps and dab them with tiny amounts of leather oil, if I want them more supple.
I you do oil a strap, do not go overboard with the amount of oil, but go easy on the leather oil.
Not that I oil that many straps. I like to wear the straps to soften them.
Just wear the strap and it will conform over time.
Once it takes the proper shape, you might prefer that to a too soft (oiled) strap.