Today, Chrome is by far the dominant web browser, which gives it significant influence and weight, including in the area of user privacy. But Chrome is beholden to Google and Google is beholden to the torrents of money that pour in from intrusive Internet advertising and the associated consumer surveillance business. This means that there are limits on what Chrome will do; for instance it's probably not likely to be aggressive about not sending Referer headers. In general, Chrome's support of user privacy will always be limited and conditional.

Firefox isn't beholden to Google (at least not in the same way), but sadly its overall usage is relatively low. Firefox still matters, for various reasons, but its influence is probably more moral than concrete at this point. People may be swayed by what Firefox does, including in the area of user privacy, but with low usage they're probably not directly affected by it. Inevitably Firefox generally has to wield its remaining influence carefully, and radical moves to help user privacy don't actually help all that many people; not all that many people use Firefox, and websites probably won't change much to accommodate them.

(Such moves help me to some extent, but I'm already taking extensive steps there that go well beyond any browser's normal behavior.)

Safari definitely isn't beholden to Google, and it has enough usage to matter. Partly this is because of absolute numbers, but partly it's because Safari is the browser for what is generally an important and valuable market segment, namely iPhone users (sure, and iPads). If Safari does something and your website doesn't go along, you may have just entirely lost the iPhone market, which is generally seen as more willing to spend money (and more upscale) than Android users. This is true in general but especially true in user privacy; Apple has a brand somewhat built on that and it has less business that's affected by being strict on it (especially in the browser as opposed to apps).

If Apple decides to have Safari do something significant for user privacy, it will affect a significant number of people in a valuable market segment. My guess is that this gives it outsized influence and makes it the wild card in this area. If Safari became aggressive about not sending Referer headers, for example, it probably becomes much more likely that Chrome will grumble and follow along in some way.

(Conversely, if Safari refuses to implement some alleged 'feature', it becomes much less useful even if Chrome does implement it.)