Use power conditioner to turn off amp

Your integrated amp may have some internal relays to 'soft start' which may or may not work with the external turn on.
You should ask the manufacturer.

Many amplifiers have a 'turn on 12 volt system' which can turn on the amp from a sleep mode.
If your amp has such, that would be best.

Turning on an amp from an external switch is a big 'hit' of current rush for the switch used. That switch may not be beeefy enough to last.
The switch in the integrated is certainly strong enough to hadle what it is designed to handle.

The smaller items I see no problem usingthe power conditioner.
A partial solution is the turn off the integrated amp with it's own power switch, then turn off the power conditioner to turn off the other stuff.
Then when you want to use again, turn on the power conditioner first, then the integrated amp.
Best solution.

I really would not use a conditioner as a main switch for any amp. The inrush of current may really upset the conditioner at turn on, and I think you would be risking damaging the power conditioner's main switch.

I use my old Adcom ACE515 for my amp and everything else. It's held up pretty well for maybe 15 years or so, and has never been upset.

I own an ACE 515. The Adcom power switch does not turn on the amp (high power) sockets on it's back. They stay on all the time if it is plugged in.
The switch only turns off the other outlets, not the one for high powered items.