Honestly, zero. They’re nice, but not nice enough for me to pay what would probably be a $2,000 annual fee. The Centurion Lounges, albeit much more crowded, still have premium liquor (with a staffed bar), comparable food, and even have showers in some of the locations.
If I cared about Flagship Lounge access that much, I’d just go for BA Silver in order to get FL access in addition to the benefits that come with OW Sapphire status.