No Warder follows his Aes Sedai around and tries to enter the Hall of the Tower, or wait outside the Hall either.

And his scaring off the first Bloodknife made things worse. If he had not been present, the Bloodknife would have been captured by Egwene's wards. She could have questioned him then, and at the very least, she would have known that there was more than Mesaana to worry about.

The entire reason she came so close to dying when she was fighting Mesaana later was because Gawyn interfered the first time. He basically ended up creating a situation where he would be critical to saving her life. Which is a great pity, because otherwise maybe Egwene would have realized she could do much better than a moron like Gawyn.