I completely understand the apprehension someone may feel when hearing "cleared to land, number 3", but after working in he system for a bit, it just feels normal, like driving on the "wrong" side of the road
If you consider that 8 of the 10
busiest airports in the world are in the US, there is an advantage to the system. You contact tower, and he clears you to land, using anticipated separation. Elsewhere, the controller would have to monitor the number 1 aircraft, until it vacates the runway- this must be more work for the controller, who may have other duties (transitioning VFR aircraft, helicopters, ground frequency etc).
I imagine it results in fewer transmissions, and works quite well 99% of the time.
Listening to the BA 777 LAS fire ATC recording is interesting. The controller, having cleared EVERYONE to land, then has to unclear everyone to land and issue go around instructions. TWICE she stepped on the BA 777 pilot trying to transmit a mayday call.
In the case of the BA incident at LAS, even if the airplanes weren't cleared to land, the controller would still have had to issue GA instructions to the aircraft on final, using the same phraseology (GA, fly heading xxx, maintain xxxx ft).