I’ve tried to install DK both globally (in C:/ProgramFiles/DevKinsta) and locally for a current user (C:\Users\slaFFik\AppData\Local\Programs\DevKinsta) - no difference.
DevKinsta v1.0.1. I’m on Windows 10 Pro 20H2. I have Docker Desktop 3.1.0 installed and working.
Docker was silently (without notifications) requesting access to directories inside C:\Users\slaFFik\DevKinsta\public and ...\kinsta etc path.
So I’ve opened Docker Desktop > Settings > Resources > File Sharing and added C:\Users\slaFFik\DevKinsta to the allowed paths, restarted docker, and Kinsta finally has started itself.
This fix also means that DK0005 error description on your site is not necessarily correct.