Chrome doesn't show in list of browsers when setting default in Windows 10

After an update to both Windows 10 and Chrome (51.0.2704.103), my default browser has cleared:

and clicking the "+" button to choose the default only gives me the option of Edge and Firefox:

Does any one know why this is or what I can do to fix it?

Sometimes I have mixed results with the ModernUI default programs. Can you try opening Control Panel and using that interface?

Control Panel\Programs\Default Programs\Set Default Programs

I have had the same issue as well on different installations.

As philosophickle said in his answer, try setting it through the control panel (+1). That has worked for me.

Another option that has worked for me is to select a different browser, like Firefox, reboot and then (hopefully) Chrome will show in the list, allowing you to select it.