At my facility, we don't coordinate each airplane like that, but we have procedures that dictate how airplanes will be delivered from feeder to feeder and feeder to final controllers (sometimes we coordinate individual airplanes). If I did get you from another controller in my facility, it is true that I don't have to verify your mode C, but if you're not level, I like to hear to what altitude you're descending; mistakes or miscommunication does happen.

If you are checking in to approach from the center, "Descending to 10,000" when you're going through 12,500 requires me to verify your mode C the same way as "climbing to 5000" does on departure.