I still use a Pocket PC 2002 based device and ran into the same problem.
These are the steps I took to get WMDC to work again:
If Windows 10 is showing you the "This app isn't working correctly" notification when you try to start Windows Mobile Device center, first reinstall it as it suggests. I simply removed WMDC first and then used the 6.1 installation available from the Microsoft Download Center:
This should make the message go away, but WMDC still does not launch, either nothing happens or it will be stuck on the splash screen.
Now go to the Services applet, and look for the "Windows Mobile 2003-based device connectivity" service. Double-click it to view it's properties, choose the "Log On" tab and select "Local System account" as the log on mode.
You should now be able to start the service and WMDC will work as expected again.