Where do we start from... There are two opposing views; both lie to the far ends of each spectrum and I am about to present a centrist view, so this may either satisfy both sides, or anger both. But I will try, anyway.
First, I will note that despite differences in the approach towards this, I dare say that everybody has the same aim at the end of the day: as a scientific site with quality control, we must strive to always use correct and clear terminology. This is indisputable - I am not up for discussing the (de)merits of such an approach.
The difference, of course, lies in how much you value "correct" over "clear", or vice versa. There is again no question that in a scientific context, the phrase "amount of substance" has a completely unambiguous meaning (it refers to - well - the amount, and not the volume or mass). Furthermore, the phrase "number of moles of water" is not scientifically correct. Therefore, if correctness were the only issue here, I would fully agree with editing all instances of "number of moles of X" to "amount of X".
However, at the same time, we must be cognisant of the role that this website plays. It is not a place for peer-reviewed, academic publication, nor is it a place for only practising chemists to converse with one another. The problem, as has already been brought up (albeit somewhat less amicably), is that the word "amount" is commonly used in a lay context to refer to amount, volume, or mass. Therefore, simply dropping "amount of water" into posts without further clarification is liable to confuse lay readers, or even students who are not familiar with this terminology, which I surmise make up a good proportion of the user base. Insofar as we want to retain the custom of non-academics, we cannot directly go ahead with the action as proposed in the question.
So: in the interests of correctness, I am strongly against leaving "number of moles" in posts unedited or uncommented on. And in the interests of clarity, I am likewise strongly against editing "amount of substance" in silently; in fact, silent editing is more dangerous than leaving things as the status quo. We can grit our teeth and bear with technically incorrect terminology, but confusing students and other visitors is a sure way of alienating them.
If we are to make everybody conform to "amount of substance", then at the same time as editing, we have to be able to educate users who have not been taught the correct terminology – not in an esoteric manner, but rather in an approachable and easily understandable manner. Therefore, there is a need for a canonical Q&A on the matter, the title of which I propose to be simply "What does 'amount of substance' mean?". Such a Q&A should take care to not insist that there is only one correct usage of "amount", but should acknowledge that the scientific and lay usage of "amount" can and does differ.
This is no doubt a challenge, but also an opportunity. As this site allows academic chemists, chemistry students, and lay people to interface, it means that we not only have the knowledge to share, but also the audience to share it with.
I would suggest, therefore, to not go on an editing spree, until such a canonical post has been created. After that, yes, feel free to edit "number of moles X" to "amount of X" – with the caveat that there must, at the very least, be a hyperlink to this canonical post.