I think the default until now is to not use alarm ( for timing operations that have this option), so should be the same for stopwatch and you should not worry
If I add a wait action of 5 or more seconds and enable use-alarm then the icon appears when the action is executed and persists until the wait period has elapsed.
This happens on my Pixel 4XL.
It does not happen on my Galaxy Tab S5e.
If I recall correctly, it was happening on my old GS9 and then stopped happening.
i don't know if this helps you but the alarm icon was bothering me aswell and i just completely disabled the alarm icon in the status bar using "system UI tuner" app, i am however still seeing it on my lockscreen, which also bothers me a bit, but i don't know any way to get rid of that one.
i don't know if this helps you but the alarm icon was bothering me aswell and i just completely disabled the alarm icon in the status bar using "system UI tuner" app
FYI, System UI Tuner is a feature built in to Android, but normally hidden. However, someone created a macro that launches it by sending an intent. To install it, go to Templates, search for "tuner", and your should find a macro called "SysteUI tuner" by Lawsonator.