The thing about eyeballs is that recollections are infamously inaccurate, people will more consistently remember what annoys them and forget what goes their way. It takes more training than people realise to properly scout and analyse just based off of what you see, almost like watching a highlights reel of a player and using that to form the basis of your scouting report.
Stats likewise can also be infamous, but in this case not about accuracy but about context and the ability to apply them to meet almost any agenda. On their own they are very much a neutral party, but it takes skill to sift through the sea of stats to find the context and the relevance rather than seeing what you want to see to support your own position.
The best results would always come from a combination of the two, no one in their right mind would recruit someone based only on the stats that someone dropped on their desk without doing any scouting. Similarly no one should recruit someone based only on what someone tells them - Alex Ferguson used to go to watch players himself even after getting reports from his scouts.
