I don't know what a nut milk bag is. I just use cheesecloth to strain any nut milk or anything else that's very fine and needs to have no grit. If I used cheesecloth for straining out carrot pulp, I'd be wringing the hell out of it or leaving it overnight to drip out. With a strainer, I'd be pressing pretty hard to get as much juice as possible. If I wanted freshly extracted juice, I'd get a juicer. We used to make a green apple and cucumber juice for a dish at one job. We used a juicer even though we had plenty of blenders.
Fresh green apple juice is worth one all by itself. It's just that good, crisp, tart, and so different from anything bottled. Of course, you need to have some vitamin C powder on hand as well for it.