Sorry strap? This car is new to me. I'm generally used to a turn signal harnesses and all those functions work. Is the horn not part of that harness or is there something separate?Is the strap from the column to the dash gone or broken?
I know the car was apart so I'll checkTo ensure the column is thoroughly grounded through its slip mounts, there should be a small black wire connected between the column tube and an upper mounting support stud. Not related to horn switching.
You are seeing the 12volts on the other side of the relay primary winding, normal. What happens when you ground the contact?