Great analogy with your dog. I noticed that when I’m doing the washing up, my DMN, for some reason, takes me to ruminate negatively about my relationships, and I constantly have to return my attention to something else. No wonder I’ve come to loathe it and let my dishes pile up.
I find it can be a great tool for creativity, but needs to be directed or at least given some task to chew on; then I can close my eyes in a half nap and all kinds of interesting associations and ideas bubble up.
EDIT: that said, the default mode network should not be unjustly demonized. Its purpose is crucial for reprioritising our goals based on what is important to us at any given time, and the problem with modern living is that we never have enough idle time to ourselves, always distracted by our smartphones, and in the long run it is easy to lose sight over what drives us forward. A simple exercise, harder than it should be for most, is to be idle yet undistracted for 30 minutes. You’ll soon get into a “big picture” view of your life, what is missing, what you wish for yourself; into a kind of goal-oriented view that only kicks in in this mode.