First I hear of a stopwatch being used for pattern flying.
Agree. Nothing in the 737 FCTM about stop watches and timing. The original pattern depicted in the Boeing 737-200 FCTM had the speed at 170 knots joining downwind (Flap 5 maneuver speed for weight).
Circuit width 1.5 nm. Abeam the landing threshold gear lever down, Flap 15, speed 150 knots and landing checklist down to flaps. About 3 miles later, turn base (which coincidently was about 45 seconds) and select flaps 25 speed 140 knots. Descend when required. Turning final select landing flaps and speed Vref + 5 nil wind. Complete the Landing checklist. With an EFIS 737, if runway had an ILS, turn on the standby ILS selector which would give you glide slope guidance on early base leg before it appears on the PFD. Useful in the simulator where visual display limited over the shoulder.