I noticed a lot of newer equipment has the standby button problem. One of the reasons I picked a Schiit Vidar is it also has a signal detect. I was looking at integrateds and found only some like Audiolab were still "stupid" Many had trigger out, but not in as they arrogantly thought they were the center of your system.
If the amp only has the off and standby, then modifying it is your only choice. It obviously has the needed relay inside, so the trick is to control it with the 12V AVR trigger. Kills the warrantee of course and you do need to know what you are doing. Most companies won't give out their schematic so rather a pain.
If the amp has a "on" as well as a "standby" then a 12V DC coil AC mains rated relay can be used. I did that in my last house. My current amp is "stupid" so I use a E-bay USB relay board to short the pins of a Pyle sequencer that brings everything up in order.
\WARNING: Auto relays are NOT designed to withstand line voltage or current. 32V DC rating, not 110 AC! Very bad idea. Get a 12V coil relay with the correct safe ratings for line voltage. ( Mouser, Allied etc) If an E-I core transformer, you can use a SSR. Bad news if a toroid.