Your problem is a perplexing one. I have a beater bike that's badly out of alignment and it pulls to one side rather strongly yet its only a problem if I try to ride no hands. When my hands are on the bars its just not an issue.
Now on the other side I wouldn't get that worried about the 'close eye's and stand on one foot' test because I'm not that good at it either but it doesn't affect my balance on the bike.
You could check your wheel alignment anyway. Set the bike upside down on the handlebars and saddle, stand back a little and sight along the wheels. If they aren't perfectly in alignment with each other then this will cause the bike to pull to one side. Misalignment however shouldn't cause the bike to be dangerously unstable with your hands on the handlebars.
Maybe its a bit of a headset issue. Maybe its a bit of a balance issue. Hard to say.
Regards, Anthony