If that happens in the Welsh League then that is disappointing as it is a reasonably high standard of football
Asking the AR to just flag for throws and off sides may make sense in a youth game when the ARs are 14 year old novices but at adult level the AR has good experience and is there to assist the referee. After all that is what Law 6 says.
I have officiated fairly recently at the professional level and what you describe doesn't happen there. Each professional game I officiated in was assessed and a major part of that process is a review of the teamwork between the officials. Assessors would hammer a referee if his pre game instructions told the ARs not to call fouls or penalties
I am sure that the assessor talked about the shirt pull but if you watch the video you will see that Labs was also doing some shirt pulling. Probably less but he was definitely pulling the Bradford players shirt (Go to 7:20)
https://www.newport-county.co.uk/ifollow/latest-videos/
I am sure that this is why the AR didnt flag for the foul. Its hard to penalize when both players are guilty
IMO it would of been a soft penalty if the referee had awarded it.