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.