I have an AMD Radeon R5 340 that does not output display. The last time it worked the display was really stretched (drivers needed of course), in the process of downloading the drivers, the display goes dark. I leave it on for a few minutes because I know that happens when GPU's are updating but I decide its been a while so its probably not updating but dead. I switch the pc off and when I turn it back on there's no output again. after a couple of power cycles and the gpu not coming on at all (the fans were my indication), i give up, put it in the other PCIE slot and then it turns on but still no display, the pc does not detect it either. Right now the fans are spinning and i'm using onboard graphics. I should also mention that i use a Display port to VGA converter for my monitor. Very old stuff I am using, I know. Also, I tried it in another system and there was no display as well.
Dell Optiplex 7010 Motherboard, 8GB RAM, I5 3470
Which Dell Optiflex 7010 do you have? Mini-Tower, Desktop, SFF, Ultra SFF?
The Dell Mini-Tower comes with a 275 Watt PSU while the Desktop comes with a 250 Watt PSU.
The R5 340 GPU card is a OEM GPU card that according to TechPowerUp needs a minimum PSU of 250 Watts. But still it should have video output and boot into Windows with a slightly underpower PSU.
What version of AMD Driver are you installing?
Here is the last support AMD driver for the R5 340 from 06/2021: https://www.amd.com/en/support/graphics/amd-radeon-r5-series/amd-radeon-r5-300-series/amd-radeon-r5-...
In case the last latest AMD Driver is at fault here is R5 340 Previous AMD Driver you might want to try: https://www.amd.com/en/support/previous-drivers/graphics/amd-radeon-r5-series/amd-radeon-r5-300-seri...
Have you tried the DVI port on the GPU card to see if it has video Output? Possible the DP port went bad on the GPU card.
See if you can connect a different Monitor that uses DP directly just to eliminate the Adapter and Monitor from the equation.