Bob, You're right - those powered vents are a real 'pain' to take apart to just clean the screen!
I installed a complete new unit when we got our AS a couple of years ago, and it recently up and quit!
I started to dismantle it from the inside as your pic's show, and found that the cheapo push button switch was 'open' - a real disappointment, as we never even use that switch - it's 'ON' all the time and we always use the remote switch in the rear bath wall instead...
Instead of looking around for another switch (your toggle looks good), I just 'shorted' across the cheapo switch so it's 'ON' all the time...
When I had the screen out of ours, I turned on the fan and squirted the fan cage with some electrical contact cleaner (spray can) to remove the dirt and blow it onto the roof, where I then hosed it off...
That little fan really moves some air, and is a valuable appliance to keep up and running!
Ray