I also don't pick up child, especially in their first few weeks, unless genuinely hurt. It hard when a child is struggling but it is only fair to set expectation of child. Instead of picking up, I take lot of time on floor with new child, making sure interacting and being included. I have lot of cuddles without picking up new child.

However, I too agree that having household errands is not okay. If your day home is open for business, the children are your primary focus. There will be days when you get nothing done except the lunch dishes. Many of us cram these things into nap time but some days, they don't all go down together and nothing extra happens.

When work from home, in any capacity, it easy to think you can do household tasks and let children play but just like manager working from home, you being paid to work not to put household errands as priority. It just a bonus if you get couple of little household tasks achieved. View it like running an errand at lunchtime. If you worked outside the house, you would not be able to put on load of laundry or other family tasks in work hours. Just because you happen to be in your home, the same applies. I find if you have that mindset, it easier.