I had cause to file 2 complaints with BA last month for the same issue of not being allowed in the lounge despite my CX Marco Polo gold card (BAEC silver equivalent) in my hand & a BA boarding card on my phone.
Each time I filed it through ba.com & got a remarkably similar copy & paste job as above. One did also mention the trouble I had with my baggage – amazing given I’d not mentioned my bags, mainly as I was travelling with none!
Both times I was bunged 10k Avios – it seems to be the bog-standard, shut-them-up response. I’d much rather the appropriately addressed the relevant issue & resolved it to make service standard across the board, however I’m not going to turn my nose up at the Avios!