Apple removed AirDrop support for old Broadcom cards that are not supported anymore, like BCM4321CD, BCM4322MC and other old cards. Your card still uses the old 5.xx driver and since this card don't have BTLE, it's probably one of the models used by 2011 and earlier Macs dropped by Mojave and that had AirDrop support removed.
Catalina changed the way BT is supported, now it integrates AirDrop, Continuity, SideCar and others that were separate subsystems with previous macOS versions within SharingXPCServices.framework.
You need to change to a card still supported by the Brcm4360 driver.
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Btw, if I'm not wrong,
@jackluke have a patch that re-enables AirDrop for your card. Ask more info on the 10.15 supported thread:
There is no Catalina AirDrop Broadcom 4360 patch for (0x14E4, 0x8*) , minimum "supported" cards are 0x9* and higher: 0xA* , 0xB*, 0xC*, 0xD*, 0xE*, 0xF* and so on. 1584874546 Till Mojave apple allowed some kind of "AirDrop legacy" that worked also on Atheros wifi cards, from Catalina they...
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