You have what is not an uncommon problem with some civilian owned LEs. It sounds like the rifle has been assembled but the trigger has not been set up and adjusted with those parts on that rifle.
The trigger is mounted to the trigger guard. Consequently, the relative position of the trigger to the sear can be effected by the woodwork and how the guard is fitted.
Check to make sure that the trigger guard's mag well loop portion is straight and flat (it sounds like yours could be bowed) and that it sits right down into the inlet and bears along the bottom of the inletting along its length.
The front trigger guard screw must be tight and the steel bushing in the hole is correctly set in length (not too long).
You don't mention if this is how the rifle came to you. A previous owner could have assembled it with random parts and and not gone through the procedure to fit them properly and adjust.
This trigger guard mounting of the trigger was an issue in so much that Enfield actually did an update to the design on the later No.4 Mk.2 to remove the trigger from the guard and hang it from a support on the receiver to eliminate the problem of trigger pull off going out of adjustment.
Your problem can be fixed, you just need to read up on adjusting pull off. The resource is here if you do a search.
Rivets for the hand guard spring clip? You might have to make them. It is the tiny brass washers that are impossible to find.