213yum, please NEVER power up a system with no heat sink! Chances are you will fry the processor very quickly. I suspect it has nothing to do with the HS but with the processor. Please remove the processor and inspect the pins. You may have a bent or broken pin or socket. Good luck and enjoy, John.
With the CPU Cooler off the Ryzen shuts down almost immediately to prevent damage from overheating. But when you install the CPU Cooler, the Ryzen is now working allowing the motherboard to show what is wrong through the Trouble LED lights.
Do you have the CPU Power connection connect on your motherboard like from the manual:
With CPU_PWR1 not connected it probably won't boot up.
From your Manual concerning the Motherboard's Socket:
Make sure you have the CPU Cooler fan connected in the correct CPU FAN port on the motherboard:
The above is just to cover the basics in making sure the CPU and Motherboard are installed correctly.
If you have done all of this, then you possibly have a defective CPU or Motherboard. To rule out a defective CPU & Motherboard, see if you can install the CPU on another compatible motherboard. If it works then that indicates you have a defective motherboard but if it has the exact problem on the second compatible motherboard that will indicate a defective CPU.
If you believe the CPU is defective try opening a Warranty RMA with the retailer you bought it from first. IF the Retailer's Warranty is expired than open a AMD Warranty Request to see if they recommend RMAing your Processor. If you purchased the CPU separately and you are the original owner they you will have a 3 Year Warranty from AMD.
You can open a AMD WARRANTY REQUEST from here: https://www.amd.com/en/support/kb/warranty-information/rma-form