I determined that my issue was with the graphics card, an MSI RX 580 8GB Armor OC, was overheating. I installed from the AUR the Radeon Profile utility and used it to monitor the graphics card temperature when playing games.
After installing, I launched the utility in a terminal and entered the command.
The utility conveniently displays the temperature on the main screen, and updates the temp every 1 second.
Note: I manually monitored the temp by alt+tabbing between the game and the Radeon Profile program.
After about a week of monitoring, I concluded that the monitor goes blank and displays the 'No signal' message when the temperature of the graphic card reached approximately 75oC.
Because others who have this GPU report it should be stable running at temperatures between 70 and 85oC, I suspected that my graphic card needed maintenance, which I never have done to a graphics card previously.
Thankfully, MSI made it easy to remove the backplate of the graphics card, as well as the four screws that secure the heatpipe/heatsink/fans onto the GPU/pcb (printed circuit board). Disconnecting the power connector to the fans from the graphics card pcb was also easy.
I wiped away the thermal paste from the surface of the GPU, as well as, from the heat pipe. The thermal paste was not hardened; however, there were areas of the GPU not covered.
After cleaning these two surfaces (no solvent was used), I applied fresh thermal paste onto the GPU and spread it around with my finger making sure all areas was covered. I also added a bit onto the part of the heat pipe that comes in contact with the GPU.
I re-attached the heatpipe/heatsink/fan assembly by aligning the four screws to the four holes on the GPU side of the pcb and securing it with the appropriate screws, as well, as the reconnected the power to the graphics card fans.
Although the backplate that came with the MSI graphics card looks nice, I noticed that the inside of it was plastic, which is a material that I expect will not be a good thermal conductor. So, I did not re-attach the backplate and left it off.
I re-tested the system with the most demanding game that I have, which is Project Cars 2, a Windows game that requires Proton). For the first time, I was able to complete the California course using a GT car without getting the 'No signal' message on the monitor, which was great!
When playing Project Cars 2 with Proton, the graphics card temperature went up as high as 82oC without a hiccup in stability.
To see if I could prevent the graphics card temperatures a bit more, I added passive cooling onto the backside of the graphics card by placing a 2"x2" and a 3"x2" heatsink (one is copper and the other aluminum) I had in my computer parts drawer. Both of these heatsinks have smooth bottoms that I could place a thermal pad between the heatsink and the hottest areas of the pcb.
This time, the radeon-profile reported the graphics card temperature did not exceed 65oC when playing the same California circuit in Project Cars 2 with Proton.
I did a bit more testing in two other Linux native games that I could not play previously due to the monitor 'No signal' blanking issue, which were Metro 2033 Redux and F1 2017. This time, the graphic card temperatures were below 65oC and I had no issues with monitor blanking or stability.
So, it seems the cause of my issue was due to the thermal paste in my ageing graphics card was no longer effective, and replacing it seems to fix it.
I am not used to doing maintenance on a graphics card; however, it seems that some GPU chips can run hot and the thermal paste may need replacing after several years of use.
For now, I will mark this past as [SOLVED], but will update it if things change.