How do I determine whether a switch can handle the power/current in a circuit?
Current and voltage ratings do not neccessarily apply at the same time and in a single operating point.
A switch is a pretty good example for this:

it must not conduct a current higher than the rated current. This is relevant when the switch is closed.

a voltage higher than the rating must not occur between the switches contacts. This is relevant when the switch is open.


The current/voltage ratings may by no means be understood as a power rating and actually there is either voltage accross the switch (when it is open) or current through the switch (when it is closed). It's not possible to have both conditions at the same time and of course, the switch would never be able to survive a power dissipation of hundreds of watts.