It's interesting to read how many different set ups schools have. And how rigid and agitated some MNs get at when they are only narrowly understanding it from their specific school set up & environment.
We live in large residential are, it's very normal in our specific area schools set up for KS2 upwards (age 8) to wait outside school without parents or formal supervision for up 5-15 mins, to wait for doors to open or after school waiting for a lift . Many parents will wait too but half DC would be on their own.
So YANBU OP, plenty of parents do this and it's a no-brainer when you have another child at different school to drive to. Fortunate it's only 10 mins early to make both journeys!
OP did not say anything about playground or expecting school staff to supervise.
At our school Only KS1 have access to or go through playground for drop off and pick up. It's locked and only opened for KS1 (7& under) secure drop offs and must have parent to see them in and out.
KS2s are different they wait in area that's outside school (not a playground but by front of school not on a road, has no car access). Many KS2s here walk to /from school without parents (with friends ) plenty of walkways and cycleways here. There is an overflow park and drop 7 mins walk away - children cross a road 😮😊 (albeit a pedestrian crossing)
I laughed at the "you must use breakfast club for 10 mins OP!!"...
Breakfast club places are very scarce at our school & several other local schools with long waiting list and you give up a place (not ask for one!) if you only need 10 mins at age KS2 upwards ! Our breakfast club is for those needing 7:15-8am start (so 1&1/2 hours to 45 mins use at least that it provides!!) as your child can simply stand outside waiting for door to open. (Not as many mums on school hour contracts here. )
It's interesting the difference set ups isn't it and it's about what school set up is for OP and her perception of risks.