I tried "Download Air Display Host Software" at getairdisplay.com and after installing and rebooting, my VNC graphics problems were different, but still there. Things were slow and the mouse had a trail of duplicate mouse pointers. Display preferences didn't show any airplay devices available so I didn't see how to do anything additional to activate the driver.
Then I decided to connect my mini-displayport to VGA dongle to the VGA connector on my main monitor. Despite having the monitor connected to another computer via its DVI port and showing video from that other computer, the Mac still sees it and identifies the monitor by name in display preferences! Connecting the monitor like this fixes all VNC graphics problems.
To anyone who can't get the hacked dongle to work or use a spare port on a monitor to fix the problem, there's also a $60 dongle called "Gefen DVI Detective N" on Amazon that you can program to repeat the signal from whatever the last monitor you hooked it up to such that anything you connect the dongle to will believe the dongle is that last monitor.
I've also read that Mavericks improves this problem but it's not clear from what I've read if it solves it entirely (certainly not for everyone). I tried Mavericks briefly on a different machine but it had too many problems like broken file sharing to Windows and inability to mount a time machine disk automatically after waking from sleep, so I don't want to use it on the mac mini.