No. See @antonio 's suggestion. My old PreSonus 1810c got replaced by an RME Fireface UCX II because I needed a multi-client ASIO driver on my main system. The Scarlett’s ASIO driver is also multi-client (and with the latest update finally includes a native ARM ASIO driver!), or I’d have just used the 1810c on my new Surface Pro 11 Snapdragon Elite mobile rig.
Is bus power an option with it? Probably not with more than one mic input, but if it is that gives exactly what OP wants.
edit: looks like it is an option - if it is bus powered and the power stays on then yes there should be a bios setting to disable USB bus power when shut down, potentially listed as a “charging” feature
The 4i4 runs hot. That means power consumption. I run my Surface Pro off of its battery. The 4i4 would kill its battery life. I could plug the Surface Pro into its proprietary charger or use the 4i4’s USB-C power brick. For me, it’s one or the other.
I just disconnect my scarlet from my macbook when I am not using the scarlet and that turns off the power to the scarlet. But, I must be misunderstanding your question?
If it’s bus powered, it might be as simple as plugging the Scarlett into a different usb port that is on a different bus that doesnt have PD / always on power.
If you’re using a power adapter, it will stay on unless you unplug the adapter. If connected solely to your PC and powered by the USB port, it should probably turn off if you shut down your PC.
Maybe it’s linked to use it as a stand-alone device