If I just hook up the old horn wires and turn on ignition, hit
horn button, I hear the relay clicking.. Is the relay screwed?
Just so I understand exactly what you're saying here - for this scenario, you don't have the actual (new) horn connected to 87, but only the OEM horn wires to 86 & 85;
When you turn on ignition, relay does not click immediately, but does when you press/release the horn button?
If I have transcribed that accurately, then the 'control' circuit of the relay is working normally.
The most likely issue (once you connect the battery) is that you have the horn connected to 87a terminal (which is the one in the middle) - that is the normally-closed* terminal.
The simple test there is that when you press the horn button, the horn STOPS.
i.e. turn on ignition, hold horn button and then touch 30 wire to battery; then release horn button;
If horn does not sound immediately, but does when you release the button, that confirms you are using the 87a terminal.
Fix is to simply swap to the 87 terminal.
* 5-pin relays have both a normally-closed (87a) and normally-open (87) terminal.
The terms are descriptive of how they operate:
The normally-closed terminal is connected to the 30 terminal when the relay is NOT energized and that circuit 'breaks' (opens) when the relay IS energized.
The normally-open terminal is un-connected to the 30 terminal when the relay is NOT energized, but 'makes' (closes) the connection when the relay IS energized.
Example of these different applications (actually within the same common component):
The OEM starter relay - that uses the normally-closed contact to send current to the headlights; when the starter button is operated the relay energizes and simultaneously 'opens' the connection to the headlights and 'closes' the contact which sends current to the starter solenoid.
Triumphs have always used this feature to remove the load of the headlamps to improve the current available for starting.