I think you'll find that the bikes requirements for oil really don't care what oil you use. There'll likely be good and bad points to all and you just have to decide what you'll put up with.

If you are willing to put up with any of the drawbacks to others for that oil, then it's fine. Nothing on a bike turns at the speeds engines that use the same grade bearings. Nor does anything on a bike create the heat to destroy the surfaces needing any special lubrication that does a combustion engine. Or heat up from the efficiency loss of power transfer or simply from friction.

However I too think that mostly the only part that needs oil is the chain. And that doesn't have to be petroleum based products.

As mentioned by another, when you get to putting oil on cables and other things on a bike, you are already in need of new soon. Especially if you oil them before they show that need. If we are just talking about wiping some oil as a preservative on cables where they run bare or other places, then not so much a issue than if you drip oil into the cable housing. But with modern cables and housings you shouldn't need too ever. At least for the common environmental conditions bikes are operated in. Perhaps on a old bike with regular steel cables that rust you might eek out some more life from them.