Are you chasing a fad or a market?
gabrielweinberg.com
gabrielweinberg.com
There are plenty of places that you can choose your competition wisely. I would rather compete against a ma-and-pa than a fleet of the brightest engineers battling for space in the new fad market.
I guess it depends somewhat on if you are looking for f-you type money or something else. If I were to compete in a hard market, I would definitely go with the let the market move to me approach.
I can see why it wouldn't be as attractive as knowing everything about the user and marketing towards them but I don't see why it couldn't work, and it isn't betraying the users trust so long as the data isn't matched to any IP or pattern.
However, my thought was if that (privacy) was not the main goal and something comes along, e.g., $$$$/partners, that could catapult you to become a market leader/giant in your core goal area then given the principle of most startups to focus on a core competency and to do it really well, the secondary goal (privacy) could be compromised.
EDIT: I don't have a list but I'm sure there have been cases where companies tweak their policies once they gain traction.