Circling is a visual manoeuvre. Why would you need to use a stopwatch? Because management does not trust you. And then they have to come up with some clever way of formalising and regulating your work. That's why the timing is the way it is ;-) Stopwatch circling is Taylorism defined.
B737. Average circle ground speed F15 is in the order of 160kts. The downwind timing is adjusted for tail winds. I find 3sec/1000' works out fine, in the sim. Considering the allowed visibility, 2400m, for CAT C a/c, and considering a foggy night, rain, low cloud etc. etc. it is wise not to venture to far away from the runway. It would be easy to stray outside the visible segment. 4.2nm = >6000m. The required vis is not related to the circling height, unlike DA/VIS on a straight in approach. True, it is a visual manoeuvre, but it is also a combination of 'dead-reckoning'. At low level in marginal conditions it is a good confidence to have a method, backed up by Mk.1 eyeball, that puts you in the right place on base leg to make a stabilised finals.
Imagine you were flying on the outside of the circuit at 500'. i.e. a right hand circuit from the left seat. The cross-wind leg is flown on timing to give you the correct spacing. You will likely be blind to the runway. Turning downwind you acquire it again. If the timing has been correct you should be spaced correctly. A pure visual judgement/perspective at 500' is not easy as it is uncommon. Passing the threshold you will once again be blind to runway. Are you suggesting that the low experience F/O doing this for their first time visually navigates you round a low level circuit? Not for me, thank you. I've experienced a dead-reckoning system hat works and I'll back it up with Mk.1 eyeball.
Remember this manoeuvre/procedure needs to work in manual flight as well as autopilot.
I once did some sim checks on F/O's from a 'gone bust' Mediterranean airline. Perhaps they were used to good weather, perhaps they had not done circling; either way they did not have a procedure. I gave them a circle at 900' at a major flat ground airfield, in 8km vis. They became completely lost downwind, starting turning searching for the runway, descended and would have crashed. They had no idea or structure of how to fly this manoeuvre. It is true that the numbers game does work for those who have nothing else.
It was proven to me at night in a caribbean island B767. Minimal lighting, no DME, no direct approach aid. It worked in real life too.