The problem is not about doing two things at the same time (enforcing rules and playing); as you said it yourself, it is not all that complicated.
However, and as you can easily imagine, the main issue with having someone referring their own match lays in the potential bias coming with the decisions taken, especially in the case of a dispute. The definition of a referee is that he is supposed to be the most neutral possible while carrying on with his task. If you are playing in a competitive contest and acting as a referee, your neutrality can easily come into question, and this no matter who you are.
The issue coming with bias isn't even necessarily always 'one-sided'. You can also end up taking the wrong decision due to being pressured by the other team questioning your impartiality. I will quote you the issue I brought up during the 5v5 hosted by the 2Lr:
Next time it would be good if a referee which isn't from the regiments playing could show up. When you have to judge a really difficult situation at 5-4 in the match as a referee and a member of the team playing, your impartiality is always gonna be questioned by the other team even if you are right. Such a situation would and will be avoided with a neutral referee.
Obviously, in the unlikely scenario of an official match having properly been scheduled by both regiments (posted within the scheduling deadline) and not a single referee or admin being able to take it, we could allow that match to be referred by a member of a regiment playing, given their opponent agrees with it. But quite frankly, with only four official matches having to be played each week and already 11 referees application having been posted (without including the tournament admins), I highly doubt this will happen.