YES it is camera shake. But ALWAYS IN THE SAME DIRECTION??.
You can't be sure of that without well-controlled tests: the eye can be fooled when there are multiple sources of distortion.
The birds and BG seem to be 'different' images.
If this doesn't happen with all your lenses, then it's very unlikely to be a sensor or processing problem. It would be weird, VERY weird for Canon to have their firmware record the raw image differently based on type of lens. Sure, it's possible that JPEG processing might take into account the lens characteristics, but not the raw image.
The problem has NEVER occurred with nearly 1000 pics using the same lens on the 5D Mk II.
There may be other usage differences between the two that account for that. Can you do controlled tests side-by-side? Are you using a faster shutter to account for the crop "zoom" factor of the 70D?
The main reason for posting is indeed 'Has anyone else experienced the problem?
So far the answer seems to be negative, so it seems unlikely to be a design flaw of the 70D model.
It's possible there are multiple issues combining for this apparent effect, try to isolate one variable at a time. Also, could you post your image settings? Also, try shooting at 1/2000th.
I'd be interested to see the results of your tests.