I started out in 1972 using an Outers cleaning kit with their solvent, oil, and moly lube, patches, and an aluminum rod with a brass bore brush and nylon jags, all packed into the usual hinged flat tin box.
It worked fine, but I switched to Hoppe's #9 and light machine oil, because the oil was easy to get and I liked the smell of the Hoppe's
I've never seen any reason to switch. Guns are still guns and these products do an excellent job.
Ballistol might work okay, but I think that faith placed in a single do-all product is akin to buying snake oil.