I agree with not soaking. It works for some, and I used to do it 20+ years ago… but I also detest using bridles that were soaked and over oiled, as they often feel too soft and worse - gummy/tacky/oily.
I find that I can get mine to the right level of pliability without the buildup by not just successively oiling and conditioning over time, but by using them. I oil and condition strap goods just enough to use them comfortably, then through use and gradual conditioning get them to where I want them over time. They break in quite nicely.
For initial series of oil for strap goods I rag or paint on neatsfoot and then roll and massage gently in my warm hands, let sit for 30m to an hour, then wipe off. Repeat a few times over the course of 1-2 days, and then let sit a few days in a warm place to let any residual oil absorb. Wipe again amd then condition. Once soft enough, I use the tack awhile, cleaning after each ride. In the beginning I like to use a conditioning soap like Belvoir with some regularity (daily/every other day), but then once tack is nice and supple, I switch to wiping with a damp rag most days and then clean/condition every few weeks and only oil as needed.
NOTE: I repeat the oil/condition/use cycle a few times, as necessary.