Ballistol is basically highly refined mineral oil, what today is called pharmaceutical-grade as in baby oil, with a couple of extras added in minor amounts (mainly to make it smell nice!). It is an interesting exercise to look up the MSDS for various commercial leather-care products, many of them use the term "natural products" but then use mineral oil (I guess it depends how you define "natural" ).
I'm guessing that for these manufacturers it's easier and cheaper to get bulk supplies of this oil than to source large quantities of neatsfoot oil, mink oil etc, plus it would be easier to get the soft consistency they want for their products. I found one product that, once I chased up the MSDS, appeared to be using hydraulic fluid, which is also a highly refined oil available in various viscosities. This, by the way, is also what is used for Lily White sewing machine oil because supposedly it doesn't stain material being sewn.