I had "school run mum friends" I felt great about it, until one DH of the mum fell out with my partner, and dragged me along for the ride.
Next thing I know the other people in this group turned thier backs on me too.
For a few months I couldn't face the school run and sent DP to do it... until one day I though f- this. My son needs me and nobody should stop me.
I walked in and mum in question was walking towards me, face like thunder, i happily said "hello X" with a smile and continued to walk... she looks totally shell shocked and said hello back.
Now I tend to go to school a little later to avoid the standing in the playground bit, and when paths cross with these mums I say a cheerily hello. Dont always get a response, but I've learned not to care.
I must admit I'm happier not doing the school run though.
Try not to let the school run become a place of fear, you're doing it for your DC at the end of the day. Nobody else. Nor always easy, because I even get the fear, but I try to overcome best I can.
Not to mention, if your feeling lonely and scared, there's other mums feeling exactly the same too. You're not alone, and also - your missing nothing not being "friends" with playground mums.
Social distancing will now be the perfect excuse not to bother with any of this stuff 