The time constant circuit in the relay sounds bad. May be easily repairable by any electronics type. Since you never hear it click off with the key on it is on.

The voltage drop across the fuse shows full current is flowing. I thought for a moment someone may have installed a extended glow type relay. Again since you never have it go off it should not be that.

Although if it were a checked only after a minute may not be enough time for it to drop out. I suspect a leaky or shorted capacitor in the relay. Or really off value resistor. It is not a welded contact issue as you hear the relay drop out when the key is turned off. That alone proves it is the problem.

Junkyards can be your friend. I have never had to service one of these relays or tried to. Just always swapped them out. This is a case or defect where I might look inside the relay box. I assume they use a simple time constant circuit with some type of temperature type thermistor to control glow duration but do not know. It could be far more complex but does not need to be.