I would start by trying to figure out why normal suspend is not working - I don’t believe it is correct to say that the CPU doesn’t support it. More likely is that there is a bug somewhere.
First thing I suggest confirming is that the firmware is up to date, as buggy firmware can cause issues with power management features.
Also, I see that your system has a NVIDIA GPU - did you install the NVIDIA driver? If so, how did you do it? Did you use the instructions at RPM Fusion?
Lastly, having more details could help with troubleshooting. Details from a tool like inxi or fpaste would help: What is fpaste and how can I use it?
p.s. interesting choice of bold formatting.