But it seems many startups go broad to start (Reddit, Mint.com, Google) and have succeeded, so how would a startup founder decide whether to go broad with their design vs targeting a particular user segment?
It's more valuable to have unbridled enthusiasm from a few users, than a bunch of users who don't care, and would easily swap you out for an alternative. Those enthusiastic users will be your champion, your marketing, your calling card and perhaps your best sales presentation. I think that is straightforward.
Using your Google example, their focus, initially was very narrow - search. Anyone might need it, but if you do that one thing really well, they'll love it and other people will hear it and have to try it.
It sounds like you are twisting up model, focus, opportunity and value.
I first heard about Google from my 1st grade teacher. He bought us 1984 Macintoshs (with his own money) to use since our school didn't see the point of buying new computers to teach first graders computer skills in 1999 (the irony is not lost on me...). When he saw me using Alta Vista one day in our school's computer lab, he stopped me mid-search, logged onto Google and the rest is history.
My teacher was kind of a hippie and had spent time on an Indian Reservation as a photographer. He was obsessed with astronomy and took us on field trips to the Griffith Observatory at night where attendance was extra credit. I don't know if this is representative of Google's initial user base but it wouldn't surprise me if it was.
TL;DR: It's almost impossible to tell in hindsight who the few users were especially when a company is ubiquitous. Needless to say, it seems to be the best way to succeed
The focus for either is equally refined, search, or books, respectively.
For instance, my mother in law is into Oprah Winfrey and might get that itch scratched by something a bit like Reddit but that is appealing to her. Maybe Pinterest is closer to that.