Because the relationship between the royal family and the media is as precarious as it is specific. The BRF has to work with them and developed a system that mostly works - after all, they need the media much more than the media needs them. Of course there are tabloid publications and gossip rags (like Daily Mail, Tatler and few others) that are below any standards, just looking for the most sensational headline possible, creating stories and blowing things out of proportions. They don't get the BRF cooperation and are just mostly ignored. But since the press is independent and free pretty much nothing can be done about that.
There are layers to this situation that one needs to consider. First of all, if the BRF reacted to every false story, every false accusation, they would have to react to ALL OF THEM. And that not only would require a huge team of people dedicated to doing just that (and with that go salaries and offices and other costs), but also could destroy the relationship with the media that are needed. The other thing, it would start a discussion "should BRF have influence over the media", that could, possibly, end very badly for the BRF. The monarchy only exist because people, citizens of UK, want it to exist. And the media are, imo, the most important instrument in creating a relationship between the subjects and the monarchy. Also, the BRF are not ordinary citizens. Yes, the monarchy is constitutional and the role of the monarch mostly ceremonial, but they're still seen as people with influence and public servants paid by the taxpayers, which means that they're held to a higher standards. The rules by which they play are not the celebrity kind (and I think this is what a certain group of non-UK folks don't get) and yes, they are criticised more harshly when they make a mistake. Is it fair? Well, not really, but life rarely is.
There's also the fact that their work defends itself. It's not only the people who marry into the BRF, but everyone - not many remember or even heard about how awful the press could be to Princess Anne. But she just went about her life and her work, not engaging in any press battles, earning the respect and love of the public. She was honest, true to herself and yet one might notice she got the least amount of grieve out of the divorced siblings. Prince Edward and Sophie went through a storm and a half at the begining of their marriage and look at them now. Prince Charles... well, we all know what happened there.
And to answer your question why are they supposed to take it - because it works. It's literally as simple as that. They just have to get through the worst, prove their worth and the press will get better with time and good work on their side. They can't fight every bad article, they can't just stop working with journalist, because that immediately invites the "is the monarchy above criticism" discussion. It doesn't matter that they cost us like a pound
As for the Tatler article, I think it was a one time thing that won't go noticed by the public too much and that it won't be picked up by other publications. Royal watchers, sure, the crowd that loves to criticise Cambridges and blames them for everything, sure. But the public likes William and Catherine, so I doubt it will do them any harm. People can see the good work they're doing, how well they're representing the Queen and the country. Not to mention Tatler is a very specific gossip rag and people there still can't get over the fact that Catherine comes from a middle-class family, but it only resonates with a certain group of people from mostly aristocratic circles.
You are asking a very good question. Unfortunately it's not just limited to the BRF as I've seen similar stories written about most of the European, Asian and Middle Eastern women who marry into a royal family.
