As a server Plex is meant to be always running, a client trying to pull files will not wake it up.
First would be to uncheck “Empty trash automatically after every scan” under server settings >Library.
If the media files are on a NAS that would also add to the complication. Depending on you usage you could switch to a low power NUC that can stay on.
Wait, I didnt try to wake up from client. When I access from client, the PC has been reloaded (accessing to Windows and so on, than open the lg app later).