I worked at an elementary school in a different district for a long time, and we had at least 1 teacher outside with the safety patrol kids (I was in the rotation to be the teacher on duty for a few years). The safety patrol stood at the drop-off spots, which were past the front entrance so the cars could pull farther in. The students opened the doors on the curb side and then closed the doors. Those 4 unloaded cars would leave, and then 4 more pull up into those spaces.
That was the "smooth" operation, and it worked with 90% of parents. Some parents, even with a teacher looking them in the eye and waving them down to the first spot, insisted on stopping right in front of the door so their child wouldn't have to walk any farther. Some would pull up to a safety patrol student but then stop short or go farther so the student didn't open the door (I never did figure out if they just didn't want their car touched or what?). And then some parents would get in a hurry and drop their kids farther back in the line, thus causing the double-lane traffic, because they would pull out at the same time as the 3 or 4 who pulled up to the safety patrol kids.
I will say that our kids were never outside doing safety patrol on their own, and some parents still didn't "follow the rules", but most did, and it was usually fairly smooth. We didn't deal with the teacher lot being in the same path, though, so that made our situation a little easier. And rainy days, all bets are off!
Good luck talking with admin about it, and keep trying. Most schools put out their duty schedule for the teachers at the beginning of the year and there is a rotation so that things are fair. You may have a hard time getting anything added for this school year, but if you can make a solid case, you might see a change for next year. If your PTA is active and loved in your school, they can make a big change too, if they ask as a group. Here's to happy drop-offs!