The way things are seeded in the last 16 now, with typically 5 or 6 lower league non-seeds every year, you're more likely to draw one (unless your us, of course).
Even this year there was a 50/50 chance of matching up with a lower league team.
(I agree that it would be easy to fix if you wanted to).
I wouldn't need much convincing, however, that the warm and cold balls was happening in the 70s.