Do you need to care if Apple changes its system font (Spoiler: Nah.)
The amount of energy being expended on this is quite remarkable, but does it matter, and should you care "Well, a little bit" and "somewhat" are the correct answers, though they don't make for a sparkling copy.
First, the case for caring. Though many people will tell you they can't tell the difference between fonts, that's usually just when someone smugly shows them a capital R and demands they say whether it's Arial or Helvetica. (Actually, that's an easy one!)
Although you might not be able to distinguish font families on demand, the font used in an operating system is so pervasive that it quite definitely gives the whole thing a distinct feel--a particular flavor. One of the criticisms of Apple's switch to Helvetica Neue with Yosemite was that after a long tradition of using highly characterful typefaces for the system font, in plumping for Helvetica they picked a font that kinda has no personality. That's not quite accurate, of course. Helvetica has a personality, it's just that its personality is--or has become defined as being--bland and unassuming.
The other big criticism of Helvetica Neue is that that family of fonts--descended from the ur-sans serif, Akzidenz-Grotesk--were conceived as print fonts, not fonts for screens. That might sound like a terribly esoteric distinction, but in fact there are many ways in which letterforms are tuned and optimized for particular media.
Here's an analogy which might help. The 1938 font Bell Gothic, and to an even greater extent Bell Centennial, which replaced it in 1978--the typefaces used to print AT&T's telephone directories--feature a nifty little trick called ink traps. If you were to look at the design of the letters on the drawing board, they'd look odd, with little bits of letters cut-away, and peculiarly spindly forms. The reason that's so is because they're designed for the context and the medium they will be used on: at tiny sizes on poor-quality paper. By designing the glyphs so that the ink will spread on the paper into those cut-aways--rather than splurging out of those intersections and creating soft, unreadable shapes had the font not incorporated ink traps--you get clean, crisp, legible text.
Now, you might think: Well, Helvetica might have been designed for print not screens, but San Francisco was designed for the tiny Apple Watch, not the bigger screens of the iPhone or Mac, so Apple is just making the same mistake again. But there's room for hope. Even as it stands, there are two variants of San Francisco, specifically designed to do different jobs in different contexts on the Watch, and it's not just the shape of the letters that varies, but details such inter-character spacing.
So it might be that Apple already had an eye on OS X when designing these weights, or even if it was determined that the optimizations made for the Watch's font weren't suitable for OS X, it's possible further variants were or are being developed to fit alongside them, just like how traditional fonts might include a display and a small caps weight, say. (That's one reason to be wary of the mock-ups doing the rounds, or of drawing too firm a conclusion from those who've already tried replacing Helvetica Neue with San Francisco on OS X. The other is that letting nondesigners muck about with fonts can often result in...suboptimal typesetting.)
There's talk of this move harmonizing the visual appearance across the Mac, iOS and the Watch, but I don't think that's a primary concern. Harmonization is usually driven by more high-level things than choices of font or particular shades of blue, and Apple has up till now been happy for different classes of device to have specific identities. (That said, do note that the new MacBook's keyboard uses San Francisco for its keycaps, rather than VAG.) There's talk too of this being a bold new era for Apple, with the font marking an inflection point in its history, but I think that rather oversells the importance of a font.
Ultimately, I'm cautiously optimistic that the rumor may be true, not in particular about the specific font we're talking about, though I happen to think it might work well, but because the reasons for choosing it--that it's finely tuned, made-for-purpose, highly considered, and, crucially, designed in-house rather than bought--might be a little signal that after what I think are a couple of style-over-substance missteps in UI aesthetics over the past handful of years, Apple might be remembering what good design practice means.
All of which, though, brings me to this point: it doesn't really, really, really matter, even within our little Apple world. It's fun to spitball about--have at it in the comments!--but it doesn't warrant the kind of vociferous knee-jerkism I've seen around the web. The best thing any of us can do is sit and wait and see what happens and then form an opinion. Then express it politely and graciously.
In fact, typeset that on a nice picture of a beach and you've got a pretty good mantra for life. Just don't use Comic Sans.