1) Victor didn’t sacrifice his whole career ‘just’ because a stranger had potential. He knew Yuri. Had probably started liking him, and the video only made it clear Yuri had a potential as skater. 

2) It also explains why Victor was so flirty with Yuri from the start: he’d remembered how Yuri had been that night and already liked him. It wasn’t without feeling, or as a joke. 

3)Oh, and what was the theme of the program of Victor’s that Yuri performed? Asking someone to stay by their side. When Victor saw that video, he didn’t just see Yuri’s skill, but also as if Yuri was calling to him to be his coach again, but through skating.