If the requirement is to have the RGB strip turn on at the same time the computer is turned on then there must be a connection of some sort to the PC.
Otherwise how will the RGB strip "know" when the computer is turned on?
As I understand the plan, you would need to have the wall outlet provide power to both the computer's PSU and the new AC adapter servng the RGB strips.
I.e., one switch.
However, that all is likely to bypass the normal power on and shutdown procedures that occur when the PC's power on button is pressed. A computer's power should not be controlled via the PSU's switch or some other in-line power switch. Bad enough to need to do that when the system crashes or freezes.
That all said there are USB controlled power strips. For example (not a recommendation nor endorsement):
https://www.amazon.com/Kasa-Smart-P...words=smart+power+strip&qid=1629719410&sr=8-4
Search/google "usb controlled power strip" for additional devices.
Read the reviews, User Guides, etc. before purchasing. Pay attention to all warnings and fine print.
Quality matters.
For the most part I would be wary about doing anything beyond simply turning on the wall switch. Takes only a second and there is much risk involved in having AC power find some direct route, via the RGB strips, adapters, smart power strips, etc. into the PC.
Just my thoughts on the matter.