Looking at the pics again, it seems that the switch is indeed connected to both phase and neutral.
If one of the poles is permanently closed, the router should still work, as the current can get through. Not sure if it is still up to code, though.
I guess there is another problem somewhere.
Looking at the board, this seems to be a simple TRIAC based circuit. Like the one described at the end of this article: http://www.electronics-tutorials.ws/power/triac.html. The little blue component beside the label is probably the DIAC. The TO-220 is the TRIAC.
I guess the TRIAC is burned out, since it should really have a heat sink:
The label on the router says it uses 7 A under full load. From datasheets of some randomly selected TRIACs that would fit the specs, all have a 7 W loss at 7 A. They also all have a junction to ambient thermal resistance of about 50 C°/W, and a maximal junction temperature of 125 C°. So, under full load the junction will heat up to about 350 C° over ambient temperature, about three times over the limit. Probably worse, since there isn’t much airflow. Basically, if this thing is run at more than 20-30% load for long enough, the TRIAC will burn out. Seems a likely scenario for what might have happened.
Replacing the TRIAC should be fairly simple. If there is some space for a heat sink, that would be a great improvement of this tool. Would be nice if someone would know a way to verify my theory first, though.