Sure - if he's aiming and not firing multiple rounds per turn. Are you really reading what you post?If you go five rounds--five minutes--are you going to allow the revolver character to get by on just six rounds?
In your method he goes how many rounds again - and how does that vary again?
Tracking each round he may be out in the first minute - or, very unlikely, but possible, 6 minutes. More than likely he's gonna be out around the 4th... but its an added drama and believability element for very little effort. Players rolling dice isn't exactly a clock stopper in my universe... they can check for ammo every round for all it would matter - and a useful activity while combat is being resolved for everyone.
My military examples are fairly standard military load outs. (8-45 round mags in two pouches was standard kit for the widely used Soviet RPK-74).
Regards your 'typical' Traveller - if the single shooter is carrying 30 rounds there's no need to check till at least the end of the 5th turn. However, if the PC only has a single load out in a 6 shooter - each bullet is pretty significant. I want a mechanic that handles that, yet also handles the armed for bear (that's not typical?) PC and a diverse variety of weapons with regards to ROF and ammunition - and in an unpredictable fashion.