No.. I don't think that would be related to the head gasket. I'm tempted to say that's a different beast entirely, like another power source drawing energy from the battery preventing quicker starts. Listen, I'm no mechanic, so I unfortunately I don't have the experience of having seen lots and lots of situations like this to thoroughly diagnose what issues your car is having. Your radiator is definitely spent, so I maintain that is a good starting point, and then see what else needs fixing.
For additional help, I'll notify a couple people that might have a better idea than me what's up with you car. I hope this is of some use to you.