Hammerspoon script for ensuring Sidecar is active when iPad is plugged in (macOS 10.15; see comments for versions up to macOS 13)

init.lua

hs.loadSpoon('SpoonInstall')
spoon.SpoonInstall.use_syncinstall = true
Install = spoon.SpoonInstall

log = hs.logger.new('init', 5)

-- function debugUI(msg, table)
--   log:d(msg)
--   log:d(hs.inspect(table))
-- end

function connectToSidecar(msg, results, count)
  local item

  sidecarItemFound = (count > 0)
  
  if not sidecarItemFound then
    return log:d("Can't get Sidecar connection menu item:", msg)
  end
  item = results[1]
  if item.AXMenuItemMarkChar == nil then
    -- log:d("Connecting to Sidecar...")
    item:doAXPress()
  else
    -- log:d("Closing menu - already connected to Sidecar...")
    item.AXParent:doAXCancel()
  end
end

function stopConnectToSidecarItemSearchTimer()
  if connectToSidecarItemSearchTimer then
    connectToSidecarItemSearchTimer:stop()
    connectToSidecarItemSearchTimer = nil
  end
end

function displaysMenu(msg, results, count)
  local menu, connectToSneezerItemSearch
  
  if count == 0 then
    return log:d("Can't get displays menu:", msg)
  end
  menu = results[1]
  menu:doAXPress()

  connectToSneezerItemSearch = hs.axuielement.searchCriteriaFunction({
      {attribute = 'AXRole',       value = 'AXMenuItem'},
      {attribute = 'AXIdentifier', value = '_deviceSelected:'},
      {attribute = 'AXTitle',      value = 'Sneezer'}})

  -- menu:elementSearch(debugUI, connectToSneezer,
  --                    {objectOnly = false, asTree = true, depth = 2})

  -- iPad may not appear immediately in the Displays menu
  -- wait up to 3 seconds for it to appear
  stopConnectToSidecarItemSearchTimer()
  sidecarItemFound = false
  connectToSidecarItemSearchTimer = hs.timer.doUntil(
    function()
      return sidecarItemFound
    end,
    function()
      menu:elementSearch(connectToSidecar, connectToSneezerItemSearch,
                         {count = 1, depth = 2, noCallback = true})
    end,
    0.5)
  hs.timer.doAfter(3, stopConnectToSidecarItemSearchTimer)
end

function connectSidecar()
  local suisApp, suisAX, displayMenuSearch
  
  suisApp = hs.application.find('com.apple.systemuiserver')
  suisAX = hs.axuielement.applicationElement(suisApp)

  displayMenuSearch = hs.axuielement.searchCriteriaFunction({
    {attribute = 'AXRole',        value = 'AXMenuBarItem'},
    {attribute = 'AXSubrole',     value = 'AXMenuExtra'},
    {attribute = 'AXDescription', value = '^Displays', pattern = true}
  })
  
  -- suisAX:elementSearch(debugUI, displayMenuSearch,
  --                      {objectOnly = false, asTree = true, depth = 2})

  suisAX:elementSearch(displaysMenu, displayMenuSearch,
                       {count = 1, depth = 2})
end

-- what we actually get when the iPad connects:
-- connect, disconnect, connect
-- so, don't trigger twice within half a second
connectSidecarTimer = hs.timer.delayed.new(0.5, function()
  connectSidecar()
end)

function iPadConnected(connected)
  -- log:d("iPad connected?", connected)
  if connected then
    connectSidecarTimer:start()
  else
    connectSidecarTimer:stop()
  end
end

Install:andUse(
  "USBDeviceActions",
  {
    config = {
      devices = {
        iPad = { fn = iPadConnected }
      }
    },
    start = true
  }
)

@peterhartree Sorry, I don't have macOS 14 easily accessible to test. You can try using Accessibility Inspector if you want to figure out what changed since the prior version, but instead of scripting System Settings you could also try https://github.com/Ocasio-J/SidecarLauncher — just read about it today. It uses a private API to enable/disable Sidecar and is tested on macOS 14.

Thank you @nriley. SidecarLauncher is working perfectly on Mac OS 14.2.1.