So I'm going to try replacing it with a solid state 40A relay, similar to this one. More on that next time 

So, I replaced the relay with the new solid state 40A one (~$2.50), and so far it is working well, much happier than the cheap Chinese 10A one (<$1).
Unfortunately, the 24v power supply had partially failed. Possibly because of the cheap relay? Maybe. I was only getting about 6.8v out of it, enough to run the audio amp but not the LED strip.
So I pulled the PSU apart and had a good look, nothing unusual. Checked all the diodes (OK), then checked the big and small transistors (all OK).
Finally I got out my Dick Smith ESR meter and went over all the electrolytic capacitors, and found one 47uF 50v capacitor that looked normal, but with ESR way high out of range! I pulled it and measured with DMM, was only reading ~30uF. It was obviously on the way out.
This cap is on the controller chip Vcc input. If it doesn't get charged, the IC won't turn on, and the secondary stage of the voltage conversion won't happen. Which is why I was getting 6.8v, not 24v.
I replaced it with a Rubycon cap with same specs. Now all is good again

Very happy to have been able to diagnose and fix that problem for myself. If you ever find yourself in a similar situation, I found this video very useful. This dude has a very strong accent, and an amusingly intrusive cat, but knows his stuff and is very informative:
Another issue is with shared grounds between 24v PSU, PC and IPAC via audio ground.
I was seeing pause and admin (<SHIFT>/enter) "get pressed" repeatedly... especially when playing music.
At first I thought there was a short in my wiring, checked everything I could, but both buttons are way up/back in the marquee area. I've already cable tied it in, I didn't want to rip it out needlessly. Good thing I didn't.
I am now 99% certain it is ground loop related. If I pull out the audio input from PC to the amp, the button problems magically go away.
So I ordered a 3.5mm plug-in audio ground isolator, see how that goes in 2-3 days time.