I’ve been making my own peanut and almond butter for decades. Just the nuts in the food processor and nothing else. I’ve never had the homemade stuff separate like the store bought natural peanut butters, so I’ll never go back.
With tahini, though, my trick is to turn the jar upside down when i bring it home from the store. after the oil rises to the top, I turn it right side up and then stir with the oil on the bottom. it goes so much easier and usually doesn’t separate out after that. That works for natural peanut butter too.