I've become weary of hearing this phrase. I'm not completely convinced that it was ever truly useful, but if it was, I think that it has long since outlived its usefulness.
To be clear, I sincerely believe that Stack Overflow and Stack Exchange have made the internet a better place and will probably continue to do so for the foreseeable future - but while such a thing is easy to see in aggregate, on a large scale, it's a worse-than-useless metric at the scale of an individual question/answer thread.
My problems with this phrase are:
It is completely subjective. In fact, the words "better" or "best" are widely considered a red flag in questions themselves, and in some cases the automatic filters will kick in and warn the asker that the question is probably too subjective.
Reasonable people will never agree on a clear definition of what makes the internet "better", because the internet is vast and different people use it for different purposes. More often than not, the definition of "better" changes from moment to moment and reflects little more than the author's personal preference.
On most sites, we judiciously tell members to avoid phrases like these; how, then, can we expect to be taken seriously as community leaders if our raison d'être is predicated on such a similarly vague platitude?
It has troublesome implications with regard to closing/deleting questions. The message it sends is that when we choose to close or delete a question, we are declaring that it makes the internet a worse place - or, at the very least, does not make it any better. This is offensive and unfair to the author and needlessly ramps up the level of butthurt.
On many if not most sites, we are very careful to tell people that closing/deleting a question does not necessarily mean that we think it is entirely without merit - just that it doesn't belong here, and that they are more than welcome to take another kick at the can provided that they follow the guidelines in the FAQ. It's much harder to get this message across when we supposedly have this doctrine that any content which "makes the internet better" is definitively allowed.
It's just obsolete; it's incompatible with much more current and clearly-worded essays and policies such as Good Subjective, Bad Subjective, Real Questions Have Answers, The Future of Community Wiki, Gorilla vs. Shark, The Trouble With Popularity, and more. Is it OK for people to just ignore two years worth of rich history and extensive debate, blogging, and podcasting in favour of some ancient handwavey one-liner? That seems oddly like representing a case in civil court by holding up the Ten Commandments.
Should anybody still be quoting "makes the internet a better place" as an actual, practical criteria for whether or not a question should remain open or closed? In my 2+ years of experience on Stack Overflow/Stack Exchange, I don't think I've ever actually seen it help to resolve a debate; it usually only succeeds in raising everybody's blood pressure by raising the stakes - instead of a relatively simple analysis of whether a question actually belongs on a site as defined by that site's own rules and guidelines, it escalates into a pissing match over whether or not the thread has any redeeming value and who has the burden of proof.
Can we please lay this doctrine to rest quietly alongside Community Wiki questions, the "Subjective and Argumentative" close reason, and other ideas that sounded way better in theory than they actually worked in practice?
Of course we all want to make the internet a better place, and we all probably believe that's exactly what we're doing, regardless of our actual approach; so, recognizing that, can we just declare once and for all that it's essentially just mental masturbation, and start using the more practical criteria we've been given over the last 1.5 years?