I breastfeed my two. Anywhere. I always tried to be discrete, not because I was embarrassed, simply to be considerate to others.
It is in the nature of toddlers to climb over things, run and shriek. That's toddlers. It is innate to them. In somewhere like a playgroup, that's fine. In a restaurant, it's not fine and the parent should attempt to distract to minimise the disruption. So I can't see how breastfeeding is different. Like toddler behaviour it is natural and there is nothing wrong with it. However that doesn't mean we should just think that everyone else should suck it up.
Likewise, when I have my period, someone also natural, I wouldn't change a bloody tampon in the middle of claridges.
There is nothing wrong with being considerate. Claridges weren't asking the lady to stop, they were asking her to be considerate to others. If someone had rocked up in shorts and a t shirt, they would have been told to dress more appropriately.
I find it a bit embarrassing the reaction to this. Women shrieking that it's their right. Their baby neeeeeds it. It makes my toes curl.