A friend on mine, who's machine I maintain, has a 5700XT and an R5 1600 and plays Diablo 3 as well and has no issues. So maybe the issue is elsewhere? I remember D3 being finicky at times.
If you would be so kind and answer these questions:
1) Windows version?(1803,1905, 2004 etc.)
2) How old is the install and how do you maintain your installation(is it bloated with tons of app installs)?
3) 3rd Party monitoring and peripheral tools(could be control panel for your mouse or headphones, could be Rivatuner. Stuff like that)
Simple troubleshooting steps to take(read them all before doing anything):
1) Open Powershell or Command prompt with Admin rights and run "sfc /scannow" without qoutes. This will repair most corrupt system files if you have any.
2) Remove via uninstall all Visual C++ Redists (both x86 and x64) and download and reinstall them again. Considering the annoyance of digging them all up from microsoft.com here is a link for a complete package from techpowerup.com ->
click me to go to their site<- .
3a) Download
Radeon driver 20.7.2 from AMD.com (link to actual download further down the page, you can also google the driver and pick the AMD link). I don't know how the 20.8.* behaves with 5000 series which is why I'm not linking to it.
3b) When installing them don't use DDU, just tick the Factory Reset in the driver installer and don't tick off "keep settings". Reboot when the driver is done being installed.