I can’t claim to be technically knowledgeable in that area either.
As I understand it having an ICC profile ensures the picture is seen the (potentially) the same (as possible) on every device. It records the gamma used and this is surely important because Macs use a gamma of 1 on their screens and most PC monitors use 2.5. This makes images look very bad on Windows if they came from a Mac. (and vice versa).
Not everything supports ICC profiles (esp. browsers and apps on Windows) but Safari does, and new iPads and iPhones support a wider colour space than rgb-s. So I think the balance is going to tip in favour of accurate colour eventually.
I think the situation now is like whenFirefox launched and all web masters said, ‘what’s the point IE is the top browser, that is the only one we need to support’. Now 10 years later we have several browsers supporting HTML properly with CSS etc. And IE6 is dead even in corporations.
So if we always strip the profile we will create a situation where browser designers say ‘why support ICC, no images have it anyway’, which we do not want.
So I would say test your images on several platforms and only strip if the file saving is very large, and the difference is very small.