The first local rule for REGL is quite short:
"League competition shall be under USGA stroke play rules, except as modified (Local Rules). Golfers shall not, by mutual agreement waive rules."
Ok, technically there are two rules there. The first (electing to play under USGA stroke play rules) is not trivial. We play a hybrid of stroke and match play every week in REGL. The Rules of Golf do not like that arrangement - combining stroke and match play is (I wouldn't say prohibited) highly discouraged. Selecting stroke play rules to play by is the only real way we could judge the 3 points for Team Net each week. It's also the only way we can record 9-hole scores to post against our individual handicaps, and it's the only way we can add our two scores together on each individual hole to compare to see which team won each hole. Let me attempt to illustrate how we couldn't use match play rules in REGL. I walk down the fairway and hit a ball I think is mine. Turns out, I just hit my partner's ball. In match play, that's an automatic loss of hole for me. In match play (depending on the format) my partner might be able to finish out the hole, but I'm done - no score for me. No score for me on a hole in REGL means I don't have a 9-hole score to evaluate team net, I can't combine my score with the partner's for evaluating the hole point, and I don't have a round to change my handicap. So, that's the rationale on the first sentence.
The second sentence in the rule is actually very important for maintaining accurate handicaps and a fair field each week for the $5 team net prize. Let me see if I can illustrate that with an example. Two teams get to the teebox and kick around the idea of playing for the red tees (you know, just to see what it's like). The course is now shorter for them, they're close to driving greens and they all post their lowest league scores ever, with Team A posting a team net of 62, and Team B one stroke behind with a 63. Team A gets the $5 for the week and every member of both teams have a sobering shock two days later when all four of their handicaps drop by 2 strokes apiece. That's probably never going to happen, but how about this example: Player A has hit a ball into the pines on the corner of #13 and the ball ends up right in the roots of the last pine and the player's only option is to hit backward into the fairway and he might damage a club to do so. Player Z on the other team, feeling sorry for having to see this play out says: "Why don't you pull that out from behind the tree - I'm OK with it, and I'll probably end up in the similar situation on the next hole and you can give ME the break." Same implications regarding scores and handicaps - we should all be playing by the same rules.