To close this saga out: it ultimately took two weeks and the very kind help of another user on here to get my bag on the way.

I filed my compensation request, only one out of seven receipts was processed and file was closed.

I thought there might‘ve been a problem with the upload, reached out to the PSL, they had it reopened after confirming all seven receipts were uploaded and another three were ultimately processed by customer service and case was closed once more.

The remaining receipts were for rental items to replace what was in the bag while I waited and they amounted to over 2000 €. I had already received the „this is our last word“ email from CS, so I had pretty much given up on it, but kept sending follow ups to the PSL, describing everything that went wrong with this case.
And lo and behold, today I got an email that I will be reimbursed the maximum amount under the Montreal Convention. Not everything I incurred, but still more than I came to expect.

It took many emails, over two months and probably being Platinum, but I‘m glad it‘s mostly resolved.

Wishful thinking, but maybe KLM will learn that sending someone to find my bag with the Airtag provided is cheaper than paying out almost 2000 € in costs on a 500 € ticket.