It's likely the starter relay contacts heated (poor connection) and fused together. As for shorts, the start switch and kill switch are in series to ground the starter relay coil. Your main switch and starter circuit cutoff relay are in series from +12 to the coil. So it's very unlikely a wiring short would close the coil as you describe. I'd say this is mechanical within the relay body/armature (fused contacts).
The starter motor is wound to be somewhat small and produce high torque. Thus its duty cycle is very small. As for possible damage, that would require an inspection of the windings/commutator to see if they are burned. Hope that isn't the case.
Both our bikes (2003) have the original OEM starter relays. So far no issues with either. The Yamaha P/N for a 1997 Serow starter relay is 3AY-81940-00-00
. It's additionally qualified with (MS5D-611) not sure if that's a house number for a Japanese relay company.
I'd replace your relay with an OEM unit. I doubt the Japanese used a Chinese part as OEM. Someone may have gerrymandered the electrical box. Check the connections to the relay for over heating, burned wiring, corroded terminals, etc.
Probably any SPST 12VDC automotive relay would work providing its contacts are rated for the starter load. Bosch makes a bunch of these. As for cranking amps, I'd guess 20-30 for the starter. Truth be known, I haven't a clue what this load is. If you search on Bosch starter relays many will show up. One example is HERE
rated for 75A, 12VDC coil and SPST contact form. Good luck with your project.