LACP transmits and looks for special LACPDU packets; it is not dependent on ARP or any other protocols.
Hardware-specific features like miimon could of course be used to bypass the delay, if the overall system implementation is sensible: if miimon indicates the physical link has gone down electrically, there should be no reason to wait and see that the LACP packets have stopped arriving over that link, but the link should be immediately detectable as dead.
Remember that a link can fail in several ways: if the cable is physically cut or disconnected, miimon can detect that just fine. But if the switch on the other end of the link crashes because of a firmware bug, the port chips within the switch may keep the links up since nothing tells them to do otherwise. In such a situation, miimon will tell that everything is fine... but as the switch stops sending LACPDUs, the failure will be detectable that way.
In other words, miimon can only detect if the other end is down or unreachable electrically. But LACP can also detect if the device at the other end of the link is "alive but not sane".