The Automatic Ear Detection setting on Mac works a bit differently than on iPhone or iPad. On macOS, you won't find it directly in the Bluetooth settings panel. Instead, the option lives inside the Sound preferences. Go to System Settings, then Sound, and look for your AirPods Pro 3 in the Output section. Click the info icon or select your AirPods and you should see the Automatic Ear Detection toggle there.
Also worth noting: the fact that your AirPods are not appearing in the Keyboard and Mouse section of System Settings is completely normal. AirPods are audio devices, not input peripherals, so they only show up under Bluetooth and Sound, not in that section.
If the Sound settings don't show the option either, try disconnecting and reconnecting the AirPods through Bluetooth, or open System Settings > Bluetooth, find your AirPods Pro 3, click the info button next to them, and check if any options appear there. On macOS 15 the layout was redesigned slightly so the controls may be in a slightly different spot than you'd expect from older macOS versions.
If the option still doesn't appear after trying all of this, a quick reset of the AirPods (hold the button on the case until the light flashes amber, then re-pair) often forces the full settings menu to repopulate correctly.