I wonder, rather than re-encoding when starting on a B- or P- frame, could you preserve the entire GOP losslessly but make the earlier content "disappear" on playback by shifting PTS, or bundling several frames into a single frame's payload? Maybe it causes too many problems with players.
That's an idea that got some discussion but generally is strongly discouraged. Cutting must mean that the underlying bytes are absolutely not there in the final result, otherwise it will cause data leaks that users would very reasonably expect not to be there.
Imagine "smart cutting" a critically compromising part of a video (e.g. showing some personal info) only to later find out that the actual footage was still there, it just needed some timestamp shifting to unveil.