I might have found the answer on those beacons.
Some brands offered these 4 way static rotators.
The beacon that LASD uses is likely of this configuration. There could be more brands that offered these besides what is on Youtube, and I am by no means saying LASD 100% used these brands. It is worth further investigation.
To achieve the CA steady burn, the front facing light was likely bypassed from the flasher cycle and directly hard wired to the 12 volt main power.