I like this theory. But it's worth noting that Moody claimed that Voldemort could legilimens without making eye contact. EY seems pretty big on Conservation of Detail, so there's a good chance that this will turn out to be important. Of course, Conservation of Detail also weighs in favor of this eye-locking episode being important, so I suppose it could go either way.

Or both: perhaps sightless legilimency is handicap-inducing like wandless magic, so eye contact might be required to read a witch as powerful as Minerva.

He already had a clue in the form of the hole Harry had transfigured in the wall of the Azkaban.