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.