I had a similar problem, but did not want to delete remove the default gateway due to configuration changes.
This KB article from Microsoft with a hotfix seemed to fix the issue:
Third-party VPN client stops Internet connectivity in Windows 7 SP1 or Windows Server 2008 R2 SP1:
Nevermind, looks like that Hotfix didn't exactly fix the issue. I believe the restart fixed it temporarily.
I deleted the default gateway for Hamachi and that fixed it. Why hasn't this bug been fixed yet? Or even been acknowledged?
Another note... After removing the default gateway, if you also uncheck IPV6 TCP/IP connection within the network properties of the Hamachi connection that also helps. I then restart the software after. This appears to consistently fix the issue.
I am having some of the same problems with my computer i did the route print and got this deleted the default route and i still could not ping, so i did the route print again and got this, i then tried to set the metric, still didn't work did the route print, and got this. I saw that it could be a firewall isue and had the computer that i cant ping turn his off and i turned my firewalls off. i have no idea what is causeing the problem.