boot by pressing key on USB keyboard

It's been so long since I've used PS/2, but AFAIK USB keyboards can wake from sleep/standby but not power on, probably because USB ports are monitored while asleep/standby but not while shut down. As for the remote button I use, there are two parts. One is a 1 foot cable that runs from the motherboard headers to an expansion slot bracket and the other is a 4 foot cable that goes from that bracket to the button. I haven't tried it, but it appears that the cable on the button can be extended using a type A USB 3.0 extension cable.

You want a key or key combo that will turn on the PC? AFAIK that isn't possible. There are other options if you can't reach the power button on the case. I'm using this https://www.amazon.com/gp/product/B08GFM9GBB/. There are even key fob remotes https://www.amazon.com/SilverStone-Technology-Wireless-Computer-SST-ES02-USB-USA/dp/B0859PKPKT/ though maybe this https://www.amazon.com/Computer-LITROK-External-Motherboard-Extension/dp/B09CZCJYSL/ is closer to what you want.