There's still an issue of trust, why should I trust Duck Duck Go any more than a company that has a reputation to lose?
Quis custodiet ipsos custodes?
Quis custodiet ipsos custodes?
So you'll catch a small niche of crazies (but small in the scope of the Internet may still be pretty big) and maybe get yourself in some more people's minds as "that place to search if I ever need to run search queries that might get subpoenaed."
The solution, though, is NOT to invest lots of time, energy, and money into make people think of you as better for privacy. You're currently the only search engine that logs so little, so you've got that niche (within people who know about you). Go court some other niches and work on making your overall product better.
As an aside, there are a couple issues with companies and reputation. First is that a company might risk its reputation if the rewards are worth it. Second is that a company might care about its reputation but simply fail to hold true to it. Finally, a company might care about its reputation some of its employees might not.