I know this is a known issue that has been affecting many sellers. I was in contact with Etsy support chat today with regards to an order issue I had, and it was recommended to me that I post it here as well.
I was recently dinged on my star seller shipping stats for an order that was "missing processing time", which also means it shows as not being shipped on time. I have confirmed on the listing that it did in fact have a processing time - it was using the same shipping profile that I use for most of my listings which has not had any issues previously. The order was shipped out same-day using a label purchased on Etsy. So in no way was the order missing information, nor was it delayed in being shipped.
Unfortunately, I made a clerical error a few months ago on two orders which resulted in them appearing as not shipped on time - my mistake. I lost my star seller badge because of it, and have been working hard to bring my shipping stat back up and get my badge back. I was back into the target range, but this glitch has lowered my shipping stat below the threshold and I will no longer qualify.
This glitch has been ongoing for a long time.
It's an entirely backend glitch on etsys side, because we cannot publish a listing without processing time.