The power comes from the battery and then goes (not at all directly) to the flasher. After that, on to the switch. The signals and the console lamp split there with wires to left, right, and console. If the problem is in the console the signal would flash and the console stay dark. The voltage will be zero right at any cross connection ("short") and rise as you move back up the harness towards the battery: this is because of the wires, connection, and switch resistance. When you reach the battery terminals it will be 12V or so unless the battery is being overwhelmed (unlikely without a fire - they can easily provide enough amperage to melt harness wiring) or discharged. Since you seem to have decent voltage to the console lamp, the switch contact for the L signal is probably dirty and limiting the load current. If so, the switch may have been damaged by overheating if you left it in L for long with this happening.
It's odd that the fuse doesn't blow - probably the switch+wiring resistance and the flasher's heater resistance limit the current enough.