That capacitor is almost certainly to prevent radio interference but i'm more confused by the fact the earth connection appears to be loose? Presumably because you were dismantling things. Can you see where the cap was meant to be connected?
I'd suggest you try the Easy Start as you're considering, if it fires the next thing i would consider is condensation in the fuel tank. The symptoms you've described so far fit with that.
If it fires with Easy Start, invest in some meths. If your fuel level is 1/4 tank or below invest in :
https://www.toolstation.com/methylated-spirit/p99550
Add it all to the tank and take it for a drive (use Easy Start to get it running to start with) to circulate it through the system and remove the condensation. Inline spark testers don't always give a good impression of the strength of the spark, the fact you've had a progressively worsening situation also suggests condensation in the tank.
Good luck and let us know how you get on.