Neatsfoot oil is meant to maintain working, disposable leather in decent condition until the leather wears out.
For a long-term item like a leather sling, I strongly suggest Pecard's leather treatment. This stuff is endorsed by conservators of most all major museums as preserving leather, and doing no harm to it.
I am a collector, and learned early-on that neatsfoot oil is NOT for collectibles, nor for long-term items like slings and even good boots. I have 30-year old leather that was treated with neatsfooot that is falling apart, and similar leather that was treated with Pecard's that is entirely
GTG right now.
Besides, Pecard's does everything that neatsfoot oil is supposed to do, except destroying the leather.
YMMV, but I deal in a good bit of leather, and I NEVER use neatsfoot oil.
Disclaimer: no commercial interrest.
BTW, the now hard-to find GI, OD, NYLON sling, intended for the M-14, but also issued during the Vietnam era is probably the best sling for most users. Few know how to use it properly, but it will do for almost all sling-users. The leather slings, and their composite descendants are good slings, but they are, by virtue of their construction, limited by the increments imposed by the holes in slings. The GI NYLON slings are much more variable, as they have no holes. They are infinitely variable, because they have no pre-cut holes.