To a certain extent, yes, the camera can have an effect. A higher resolution sensor will show more blur when viewed at 100%, although when normalized for viewing size it makes no difference.
Biggest differences are mostly due to ergonomics and camera weight distribution. Larger heavier cameras tend to be easier hold steady, which is why the Canon and nikon pro cameras are big heavy bricks.
Also bear in mind the 1/focal length rule of thumb refers to the effective focal length. So on a 1.6x crop camera a 50mm prime would require 1/8th at a minimum for an average user getting an acceptable but not perfect success rate. E.g., at 1/80th a second out of 10 photos you might get 6 sharp, 3 acceptable and 1 failure. If you bump that to 1/152th you might get 7 sharp, 2 or 3 acceptable and maybe 1 failure. at 1/200 you get 8-9 sharp 1-2 acceptable and minimal chance of failure.
And that is if you are average. Poor technique of shaky hands and your odds go down so you need a faster shutter speed to maintain a reasonable success rate.
Also, if you shoot at very narrow depth of focus then you need a lot more care that focus is accurate so you might get a blurred subject simply because you moved a few mm.