When a person dies, what made them a person is suddenly gone. What remains is a lifeless shell. I never saw a dead body that looked like it was "just sleeping". It may sound callous to some, but dead people are not people. It's a little bit different in the US though, compared to the EU, where I come from. Over here, funeral homes do not go to heroic lengths in order to cosmetically make a dead body appear lifelike again. But overall I stand by my opinion that it takes an extraordinary effort to look at the dead and not see that the person who lived there is not inside anymore.
Contrast this to the article, where dead people are being talked about as if they were still alive. I suspect these wishes of taking care of dead bodies of loved ones comes from an inability to recognize what death actually is. To me, passages of this article read as if the dead somehow required nursing and comfort.
There are many good reasons why it's a good idea to use the services funeral homes provide. First off, I am very suspicious that following such a huge loss, handling the dead body of a relative is a good idea psychologically. I am very grateful I did not have to do this when family members died. Even in cultures where families are expected to take care of their own dead, usually outside people from the village come in to help with the actual process.
Most importantly though, a lot of dead bodies pose a public health risk, especially those of older people with a history of hospitalization. I would be very concerned about multiresistant infections such as MRSA. It's not safe for family members to handle these bodies.