I have an older all metal wing capper. when i got back into homebrewing it was quite stiff and all I did was lube the moving parts with some light oil. It now works great.
if you do not have air at your apartment you can put water in your boil pot and put in the chiller, boil the water and evaporate the water in the chiller. You don't need a full five gallon boil just enough to cover the coiled part. then the hot water is used to clean all you other equipment...
I was hoping to brew this as I love a good British ale.
When I type this recipe in to BrewTarget it comes out as Cloying with only 14.4 IBU's.
Could you have typed something wrong with the recipe.
I did this
https://www.homebrewtalk.com/f51/drill-motor-grain-mill-314721/
And it works great.
Also with out messing with the drill I still have it if my other drill is too small for a job.
How large are your gas lines?
Can you see the gauge on the regulator and see if the pressure is dropping while tapping?
As the keg empties the gas space will increase which will help your problem.
I looked at your drawing and I wonder if the spark output is tied directly to the spark plug. On the furnaces here at work the spark output ties to a transformer which steps up the voltage and then delivers it to the plug. Also your burner controller is powered by 24VDC and most spark plug...