I know that for me that is false. I often get my best ideas on database design or solutions to sticky programming problems while filling sheep water tanks, or spinning wool while watching a simple mindless TV shows. And I don't consider the work of mucking a barn or digging a ditch to be lesser than the work of programming or research
I don't know for sure what Cal Newport is talking about, but what you're saying doesn't seem to be contradicting my thinking. What you describe above is a situation where you choose your tasks and you choose when to interrupt one task for another. And at times when your mind doesn't need to be highly engaged with the current task, it is engaged with other tasks.
When you actually sit down to implement the solution to a sticky programming problem, wouldn't it hurt at least a little if you were interrupted at least every thirteen minutes, and you were absolutely required to address the interruption for a few minutes, engaging your mind fully with the interruption, before you could return to the programming? I don't remember, offhand, how you felt about the "change tasks every seven minutes" suggestion, which I see as closely related.
9:48: You sit down to make changes to a database.
9:57: "I need you to stop what you're doing now and calculate how much feed we'll need for the next ninety days."
10:08: You're back at the database.
10:19: "I need you to stop whatever you're doing now and write two paragraphs describing the historical influences behind your latest weaving project."
10:27: You're back at the database.
10:38: "Sorry, I got the assignment wrong. I need you to discuss the weaving from an art history point of view, and also discuss what materials were used to make the dyes. But still only two paragraphs."
And so on, and so on. My suggestion is that those interruptions are likely to make the programming less effective, and that figuring out how to move them, so that you can have a block of programming and a block of knocking off small tasks, would be more efficient and effective. Sure, there's still the chance of, "Emergency! A fence is down and sheep are flooding into the road!" But that sort of emergency wouldn't happen every few minutes every day.
Re the deep work, I'm not seeing that as more important; I'm seeing it as not unimportant, and as work that requires undistracted focus. So my definition is circular--work that requires undistracted focus, requires undistracted focus. And some such work does exist, and some such work has value.