If I was a shareholder, booking access would be the top thing to fix. Most other airline sites I use have good to great sites (Ryanair the best, Air Europa worst; Delta good as are Chinese airlines). These AF booking issues are a huge incentive to take my business elsewhere.
I am Platinum for Life and recently tried 4 browsers on 3 different computers, and none of them work (including 2 browsers that I never used before). Android access however fine and my ID /email fine. Platinum help line wonderful and I went through all the issues and they finally said they couldn't help me. They will however waive the 30 euro booking fee.
My problem however is that I do multi long distance trips in Business and I don't like using the phone or calling for long itineraries. I am sure AF KLM are losing some of my business as I can't easily get offers.
=> shareholders wake up !!