Why does airdrop work with my Macbook, but not iMac?
I beliieve I've managed to answer this for myself.  All my previous research only turned up issues around Airdrop not working with laptops, but I think the issue is the same.

Found this thread on the Apple discussion forums that says that you need a 2012 or later iMac to use Aiirdrop.  Missed it by that much!


  AirDrop system requirements To send items to an iPhone, iPad, or iPod
  touch, or to receive items from those devices, you need a 2012 or
  later Mac model with OS X Yosemite or later, excluding the Mac Pro
  (Mid 2012).
  
  To send items to another Mac, you need:
  
  
  MacBook Pro (Late 2008) or later, excluding the MacBook Pro (17-inch, Late 2008)
  MacBook Air (Late 2010) or later
  MacBook (Late 2008) or later, excluding the white MacBook (Late 2008)
  iMac (Early 2009) or later
  Mac Mini (Mid 2010) or later
  Mac Pro (Early 2009 with AirPort Extreme card, or Mid 2010)
  


The good news is inexpensive and very small USB adapters can add BTLE and AirDrop to older Mac with macOS versions that otherwise support AirDrop. 


Asus BT400, IOGEAR GBU521 and GMYLE all seem to do the trick in several articles