Welcome to the forum.
What causes amplifiers to not come on is no electricity, perhaps caused by shorted parts inside. It is amazing what lengths people will go to avoid repairing the fault in something. The 20 amp fuse in a 5 amp holder is a good indication that a previous owner was one of those "fuse repairmen". The little box with the momentary button, the device inside is probably made by klixon, TI, P&B, or Eaton. The little button probably has an amp rating printed on it, because it is a circuit breaker. apparently rather than replace the device shorting out the power supply, the previous owner inserted a circuit breaker so he could power it up again by pushing a button.
Most common device to cause shorts in an organ with hard wired speakers is the electrolytic capacitors in the power supply, which dry up and start shorting at about 20 years. These are tall aluminum cans with circles, squares and triangles on them for the various sections, or small aluminum cans with a plastic or cardboard wrapper with a plus on one end or a minus in balls pointing at one lead. Repeated over and over repowering of a shorted amp can damage the power transformer, but if you are lucky the previous ***** only blew out a $1 rectifier or power resistor or something. Wurlitzers of the early 60's are noted for having $.09 capacitors sprinkled throughout which are prone to short out and cause various functions to stop working or overload the main power supply. I bought about 50 e-caps to start on my 4500, and intend to replace all of them before I stop. I put 71 in my Hammond, that should do the job for another 25-35 years.
I don't have a 4460 schematic diagram to give more specific advice. If you are going to try to repair it yourself, read www.aikenamps.com/safetytips.html before diving in. You'll need a dvm to prove parts are safe to touch, I prefer the non-autoranging sears 82140. If you decide to replace some parts, you'll need a soldering iron like a weller WP35, a screwdriver (not pointed) tip for it, tin-lead rosin core small diameter solder, needle nose and slip joint pliers, klein or miller wire strippers (26-16 ga) a steel pick (sears) a sharp diagonal cutters (fishing department), a 5 way screw/nutdriver. If the organ has vacuum tubes see http://www.diyaudio.com/forums/tubes...ube-amp-3.html for a tutorial on replacing the e-caps. If it is transistorized, see http://www.organforum.com/forums/sho...146#post280146
If you want a little general training on debugging transistor power supplies, pick up a copy of Electronic Devices, the Electron Flow Version by Thomas Floyd or a similar community college text.
Good luck. Some Wurlitzer organs are worth the effort to repair them and upgrade all the capacitors for another 40 years. Some are not. I don't have an opinion on this model, I don't know what it is. My 4500 and 4300 seem well worth the effort.