Is it a bad idea to drill a hole through this door frame to pass wires?
Not a bad idea what so ever. However, let me clarify on the posted answer... A hole saw can be used to cut clean perfect holes, but are limited in cut depth. You'll most likely need to use the hole saw on both sides of the jamb.

A hole hog, as another commenter mentioned, is also known as an auger bit. It has a pointed threaded tip that will pull the bit thru whatever material you're trying to drill thru until the tip can't pull anymore. From there you'll need a little extra force to get the bit to finish it's cut.

Both are feasible tools for what you're looking to do.