I've tried to reproduce the issue here on our premises. Here's the configuration:
Desktop computer with Windows 8.1 installed, including recent updates. Motherboard: Asus P5QSEPlus with "Power on by PCI devices" enabled in Bios Settings. Network card: integrated, with "Shut down WoL", "Wake on magic packet" and "Allow this device to wake the computer" enabled in the properties.
There is a wired connection to another computer via a router. Both computers are on the same subnet. Wake on LAN goes without a hitch, even from the shut down state. Once the computer is woken up, Remote Utilities can be established right away.
Please, check if Remote Utilities service starts after waking up and if anything prevents the service from operating. Maybe some security settings, permissions or antivirus software activity.
It's "either..or". Unfortunately, there's no failover mechanism to automatically switch to a default server when the GMS is down. This would add more complexity as it seems at first.
I recommend using the default server if there is a slightest possibility that the GMS PC is unavailable. We constantly monitor our mediation servers and they are located in industrial grade data centers. If they are down (which is very rare and mostly due to maintenance for a very short time) we are notified immediately and can take all measures necessary.