By way of comparison, during my last trip to London (on KLM), I got a text the day before the return advising me that the flight was full and that baggage exeptionally could be checked for free.

I arrived at the Skyteam lounge in T4 [I had no luggage to check in so didn't meet any check-in/KLM staff at all], handed my boarding pass over, and after a bit of to-and-fro between the ladies [in what sounded like Russian], they told me to head to Gate 20 and that I would be accepted on the flight that was just about to close.

So I headed down to the gate and they put me on the earlier flight.

It's surprising that any airline, facing the likelihood of bumping pax and paying compensation, would not allow such requests in this circumstance!