These systems are entirely built on code, and code is malleable. If you have control over the code that gets executed at every point in the stack (which the operator of a web app certainly does), there's always room for the operator to change the code and therefore change the behavior to do anything, including log things that were previously considered secure.
> The entire security of Lavabit depends on Ladar and his principles.
That has always been true. If he was a less scrupulous individual, he could have run an off-the-shelf e-mail solution with 0 fancy security, but claimed that it was all secure on the backend in his marketing. You wouldn't know the difference, because you can't audit the source code.
Unless you have the access, time, and expertise to perform your own source audit, any claims of security are always implicitly built upon a foundation of trust that the operator is doing what it claims.