Firefox is the browser I want to see be the competition.
Extremely unlikely unfortunately unless the planets align and Google ceases all promotion of Chrome and Mozilla dramatically improves their own promotion game. Firefox would probably need more tangible reasons for people to switch too, because realistically the masses aren't going to care about the privacy or marketshare angles.
I don't see how Mozilla could "improve their promotion game" to stand up to that.
But that alone wouldn't suffice. From what I've seen of Mozilla's efforts to promote Firefox, they don't have any idea of how to appeal to the broader audience, or really any audience beyond the privacy-conscious. That Firefox allows better control of one's data is great but it's not nearly enough to carry most people past the friction point of switching browsers. For that you need things like major advantages in battery life and features that fix common web browsing annoyances.