I don’t know if turning off Bluetooth protects against flipper attacks (Edit: Nah.), but unless something has changed, it (sadly) doesn’t preserve your privacy.
It’s not really documented, as far as I can tell, but Bluetooth low energy stays on, even when you toggle Bluetooth off for both iOS and Android. As of iOS 15, even turning off iPhones means the phone is still trackable. (Unsure about Android on that front.) Apple’s ‘Find my’ network uses Bluetooth low energy, same as Bluetooth beacons.
I commented elsewhere with an explanation and a bit of speculation. I did later confirm that even ‘disabling’ Bluetooth doesn’t stop the attack.
The attack method works even when Bluetooth has been disabled using airplane mode from the control panel, which may surprise you. In which case, you’ll be shocked to discover that disabling Bluetooth this way, erm, doesn’t. Instead, you’d need to disable it directly from your device settings or run your iPhone in Lockdown Mode to prevent these advertising pop-ups from being received. Source
Assuming similar on Android, it’s possible, but not that easy toggle everyone knows about.