This is why kids that go to child care are healthier and don't miss so many days of school those first few years. This is also why so many moms who go back to work and put their kiddos in child care or when they start school have to take off numerous days due to sick kids. They get everything, one right after the other, until they go through all that then they are still going through the stuff the other kids are bringing each day.
They build strong immune systems since they've already gone through all this by the time they get to school. By 2nd or 3rd grade they even out and have all the same immunities as the other kids from what I've read. Some say 3rd or 4th grade but I think it's closer to 2nd.
Of course we keep our kids too clean. They bathe sometimes twice per day and sometimes we don't let them outside to dig in the dirt or play in the mud. It's natural for kids to be filthy and happy. They need to eat dirt and play with worms and have fun with nature. It's how they build that immune system and get strong.
There has been research out there for years, probably over 20 years, that says kids that are kept too clean develop MS as adults. They never build the sheath that goes around their cells to make them strong so their bodies start attacking themselves and they get MS and die young or live a long life with a lot of complications.
Being too clean is bad. BUT I truly thing that you can say anything you want with your statistics. They could just as easily say everyone who has a porch light is inclined to be sicker or those who play with Legos are more inclined to get the flu. This is because everyone has those things, statistically they do. Looking at a Bell Curve there are tiny bits on each side of the bottom that are something like 10% of those counting in the research. If one says "Everyone" that implies everyone that fills the 90% in between the outside bits on a Bell Curve.
So, many many people have dishwashers. Inside that Bell Curve of everyone living in 1st world conditions probably 75% have dishwashers. There would be massive health issues going on if the article stated in your question was accurate.
They wash the dishes using hotter water than a person can stand. They might or might not dry those dishes using hotter air than that person can breathe. THAT KILLS GERMS AND DOES NOT MAKE PEOPLE SICKER. THAT KEEPS THEM FROM GETTING MANY DISEASES.
So I'd say that they were looking to show statistically people who had dishwashers are sicker than those who don't use them. They could also say people with indoor plumbing have the same statistics or people who have refrigerators have the same statistics because most people they are talking about, that they included in their study also have those things in their homes, across the board. So they generalized.
There is NOTHING wrong with washing hands and using an occasional hand sanitizer for prevention of the flu or other communicable diseases though.