There is key difference between a radar gun vs stopwatch reading. A gun locks on the peak velocity immediately after release, while a stopwatch measures average speed over the approximate 60 foot path, which would be several mph less. The current convention is to measure pitching speed at release, which only radar can do
As for the stopwatch method, even if you had Wyatt Earp using the worlds fastest watch mechanism the watch results would vary by more than 5%, or a few hundredths of a second each time.
Take a stopwatch and see how fast you can click it on and off. If you are among fastest Dads this side of Dodge City, you might barely break 0.1 second... still introducing a substantial amount of error into the 0.5 second reading.