Try it for yourself; take a hand held shot with 50mm focal length of something with lots of fine detail that's far away from the camera. Take the same shot five times at 1/250 and then see if they are all as sharp as each other.
Ok, I get your suggestion a try, or rather something lose to your suggestion. I shot a sign with lettering of different sizes from a distance of ~60 yards, with a lens set to 70mm and a shutter speed of 1/320 s. I was at f/4…that means a DoF deep enough to safely ignore focus (everything from halfway to the sign to infinity was within the DoF). I took five shots handheld. There was, indeed, a small variation in the sharpness of the shots, with three of them very slightly sharper than the other two.
But that's not the whole story. The lens I used was the 70-200mm f/2.8L IS II, and for the above tests the IS was turned off. I then turned the IS on, and took five more handheld shots. In that series, there was also a slight variation in the sharpness of the shots, and the sharpest of the IS shots was no sharper than the non-IS shots.
I then mounted the lens on a tripod, and took five more shots. This time, there was no
variation in the shots, and all of them were similar to the sharpest of the handheld sets with or without IS.
From that, I would suggest that when shooting at 1 / >4x the shutter speed, IS makes no difference, but using a tripod
is likely to get you consistently sharp images.