Do a video interview, get on several podcasts. Talk about Material design, about privacy, about android development in general. Have a blog and link to it. Personally talk to your users.
If some known person (open-source contributor or a startup founder) start using your product, ask for cross-promotion or endorsement of any kind.
Just be open. The more people know about you the more they are inclined to trust you.
People don't trust Mozilla because they read the code, they trust Mozilla because they do exactly what was described above. They do the things that foster trust.
I'm afraid Firefox is so abysmal noone could comprehend the whole project's code. Guess, we could only review a tiny bits of it (I had briefly read sync-related parts of code in hope I could replace them with something saner and simpler, but ditched the idea) and hope others did the same for other parts.
It doesn't solve the original problem as it is stated, but it still solves the original problem. 99.99999% of Firefox users will never look at its source code, they just trust the Mozilla brand.
If some known person (open-source contributor or a startup founder) start using your product, ask for cross-promotion or endorsement of any kind.
Just be open. The more people know about you the more they are inclined to trust you.