Is our flooring contractor extending our hardwood floor correctly?
I have seen glue down floors with no paper, it was a wider board, but still solid flooring, not engineered. When I was coming up in the trade when I was in my 20's I was told paper was used as a cushion, I thinks it was wrong. When nail down wood floors move as with a cushion under it, they squeak.

The paper is used to prevent I think, and you may find elsewhere, to prevent the mass migration of moisture from below to the flooring above, and perhaps to keep spills from going through below. Moisture going through a floor is inevitable and necessary. The rapid transfer is harmful and will damage the floor. The paper slows the transfer down. The floor finish and cleaning up spills asap is part of that.

Where the paper is missing at the joints can only be repaired to a degree, the driving of the new floor will have a tendency to rip the strips up as it is being driven in if they are narrow. Narrow strips will have many joints anyway so in the long narrow areas it may be a moot point. I think some areas may have been able to have been detailed better, like sliding the paper up under some of the ends to cover a little better, but if the action doing this loosens up the existing, then it is better not to do so for creating squeaks.

IMO, they are doing an somewhat acceptable job on the paper, their toothing is excellent, tying in new to old and finishing it usually stands out a bit, but spreading out the toothing long into the old floor helps blend it in. The ends of the flooring where they are driving it in, leaves a little to be desired, I see the ends are getting dinged up from the hammer by not using an effective drive block to protect the new floor while being driven.