Should they be running the whole game? The RFL has always focused on the professionals at the expense of the amateurs, which is why BARLA exists in the first place. They need checks and balances so they don't throw everything away to preserve their loss-making professional league. For instance, in Scottish football, there are seperate governing bodies for different levels of the game and their leaders sit directly on the SFA board. While there are probably too many of these governing bodies (for instance, amateur football and welfare football are practically the same and should be under one body in my view), they provide a different perspective. I honestly think they should have an amateur rugby league association that controls the game at an amateur level, leaving the professionals to the RFL. They can't do amateur rugby, they've essentially been killing it for 20 years whilst hiding this fact through Sport England money. It's not a case of historical distrust, it's a case of their historical actions.
I wouldn't use Scottish Football as a good supporting case for multiple NGBs. Most top level sports have a single controlling body - without it you will inevitably end up in conflict, plus you have issues surrounding external funding.
In normal circumstances the RFL should be running the whole game - I don't think there's any question that there should be a single NGB. However history has left rugby league in a position where the NGB is not trusted to run the game at grass roots. Very sad.