Wouldn't that depend entirely on how close they got to any individual Oort cloud objects?
Would a 10 km near miss of a 1/8 solar mass, red-hot object affect a cryogenic ice ball's path?
Yes, on an object, but not on the cloud overall. Cometary bodies at that distance are thought to be very sparse.
The answer is somewhat self-evident, however, because the solar system did not experience a massive influx of cometary bodies in the last 70k years. :)