The grub screw on the left (at the bottom of the backgear detent hole) sits in a groove in the backgear eccentric. It locates the eccentric sideways. It needs to be backed off a little to allow the eccentric to turn. The grubscrew is locked in position by a second grub screw at 90 degrees to the first and is inserted from the right.
Above the second grub screw is a third which is the one you remove to insert a pin into the wick. (I think that's the one marked 'allen key' in your photo). No need to use the pin when you strip out the spindle. You only need the pin on reassembly. Poke the wick down against the spring and hold it there out of the way with the pin until the spindle is in place. Then withdraw the pin and replace the grub screw.
Andrew M