I just caught up with this and must admit I multitask while working CW all the time.
Never really thought about it, but then I learned the code 49 years ago and learned without writing anything down, just listening to it. I still never write anything down and just listen to it, and often turn it over to the other op and copy while I walk into the kitchen and cook dinner or something, chat with the XYL and kids, and watch the TV news while listening to the contact, and really don't miss anything.
While sending with my right hand (trying to get ambidextrous, but it's not that easy) I often read the newspaper using my left hand to flip the pages, and never really thought much about that, either. Two different activities, they don't seem to interfere with each other.
I remember giving a "mobile CW" demo to the Simi Settlers ARC locally about 15 years ago. I was operating mobile CW and a whole fairly large group came out into the parking lot to watch that, and while watching it, they were asking me questions while I was sending maybe 35 wpm or so, and I was answering them. Two different activities, so they didn't interfere for me. Those guys still talk about that, as if it's some unique talent, but I don't think it really is.
Can you shift gears and change the radio station on your radio at the same time? Or shift gears and roll down the window? I'll bet you can.