No matter what sensor you use, you can never really out-resolve your lens. What I think you're seeing here is that the 300mm L II just isn't a good enough lens to allow the 7D to outperform the 5Dmk3. If you believe the DXO analysis, they rate it at just 21 MP of resolution on a full-frame camera. Assuming those numbers are correct:
1. You're only effectively getting about 8.4 MP of usable resolution out of your 7D when used with that lens.
2. The 5Dmk3 exceeds the maximum angular resolution of the lens, so no other sensor can significantly outperform it.
A 7D's sensor, if scaled up to be full-frame size, would provide a whopping 45 megapixels. Thus, to fully take advantage of the 7D's resolution, your lens would need to be capable of resolving 45 megapixels on a full-frame sensor. If a sufficiently sharp lens existed, in theory, the 7D would mop the floor with a cropped image from the 5Dmk3 when used with that lens. Unfortunately, I don't think such lenses exist yet, and probably won't until Canon decides to build a high-megapixel full-frame camera.
Of course, in practice, the DXO analysis is, IIRC, limited by the resolution of the camera used to take the measurements, and may not adequately reflect the true maximum angular resolution of the lens itself. In other words, take everything I just said with a grain of salt.