What he said, you should do.
You'll find conflicting opinions on many collection preservation issues, and the treatment of old leather is sure one of them.
I do know that there were many methods of tanning leather, and this means that treatment that works well for one old piece of leather may be bad for another which was tanned in a different way back in 1864!.
Mix in the different sorts of thread used in the stitching, and the different treatments which might have been used on this over the years, and that adds to the possibility of damage.
I've always heard that neatfoot oil will destroy old stitching. Notice that the photo above, the bottle is labelled "neatsfoot compound" and that means there are likely some other additives.
All this means that what works for one item might have different effects on another.
I think this looks okay as it is - it had likely stayed in good shape without treatement for 150 years and can last a bit longer.
((But if I was going to put something on it, and it was my leather, I would use the Pecards made for antique leather))