I am no expert on such things, so please take this as a suggestion for further research! However, AIUI the output relays on power amplifiers are included to protect the loudspeakers from switching transients - they delay connecting the output until the amplifier has stabilised. you could probably achieve the same protection by turning the volume control to zero while switching on, then turning up to listening volume after a few seconds (having, of course, by-passed the relay when it has died).
A little observation with an oscilloscope would probably show you what was going on.
I'm sure a better solution would be to find a suitable replacement relay, but I can't help with a suggested alternative. In fact, it is probably not a simple relay, but must incorporate a delay switch (thermal?).
Best of luck!
David