How to keep System Events from deactivating

The System Events process does not start at login, so the delay might be needed for some other reason. System Events also teriminates automatically if it is not used for a few minutes.

It probably won't help, but you could try launching System Events at the start of the script:

launch application "System Events"

You could also try using a launchd agent like this to launch a System Events process at login:

<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE plist PUBLIC -//Apple Computer//DTD PLIST 1.0//EN http://www.apple.com/DTDs/PropertyList-1.0.dtd>
<plist version="1.0">
<dict>
  <key>Label</key>
  <string>test</string>
  <key>Program</key>
  <string>/System/Library/CoreServices/System Events.app/Contents/MacOS/System Events</string>
  <key>RunAtLoad</key> <!-- run at login -->
  <true/>
  <key>KeepAlive</key> <!-- run the program again if the previous program terminates -->
  <true/>
</dict>
</plist>

You can normally use something like tell application "System Events" to exists process "Finder" to check if a process exists, but it obviously requires talking to System Events. In this case you might also use a shell script like this:

repeat 100 times
    do shell script "ps -eco comm= | grep -x 'System Events'; exit 0"
    if result is not "" then exit repeat
    delay 0.1
end repeat

ps -e is identical to -ax (include processes from all users and processes without a controlling terminals), -c prints command names instead of absolute paths in OS X, and -o comm= prints only command names without the header row. You can replace the shell command with pgrep -x 'System Events'; exit 0 in 10.8 and later.

Your problem is independent of the accessibility permission.

Replace your click line with something like click window 1 and it will ask you to allow or deny access to the accessibility features.

There is a problem with click at and I don't know what it is.

Edit : My conclusion after some tests : click at is broken.
The release note of AppleScript updated with the release of Mavericks does not state some kind of removal of this feature. Some that might be a bug.

I submitted a bug report to Apple.

Edit : The issue has been fixed on Yosemite, click at works well again !