Electrical - 12v but no bulb illumination?

Stupid question, but have you tried another bulb? I have had new bulbs give me issues before, generally when a policemant comes to check the car out as well!

As it's an H4 bulb and thus presumably combined main and dip, does the dipped beam work OK?

If it does, then that suggests that that the earth is fine as that's common to both main and dip on H4's, and thus logically the issue must lie with something on the high beam wiring/connector side.

If a meter tests at 12v, it's still not a guaranty that there is enough amperage coming down the wiring to light a 55w bulb. I would expect with the symptoms to find a heavily corroded terminal, faulty burned relay or switch contacts or a wire where the insulator looks ok but the actual core is broken but carries enough amps to swing a meter to 12v but not light a filament. Start by making a new earth with new wire and terminals, if that fails, you'll need to start tracing wiring, or running in new loom sections. Take this as an opportunity to build a relayed headlight loom if the vehicle doesn't have relays as standard.

leef brings up a good point.

A bad connection can leak voltage through but not enough actual amperage to power anything. Older cars (Especially American) that use disc fuses on the back of the cigarette lighter sockets do this... the fuse will blow but on a Multimeter you'll get battery voltage or darn close at the socket but forget about lighting up a test light.