Just curious, how exactly did both weapons ultimately have the same effective range? I would have to disagree but, I'm always open to learning new things about weapons.
The big issue in question is range, so we'll start with the basics of what constitutes an effective range of an LMG.
The BREN fires the .303 British, it hurls a 174gr bullet roughly 2400 fps. The BAR fires the .30-06 Springfield, it pushes a 150gr bullet roughly 2800 fps. The differences are minute enough to hardly make mention of it. Do you want to get your hand smashed by a 18lb or 20lb sledgehammer? Next, we'll go to the sights. Both featured iron sights (no optics). Both featured a "ghost ring" rear sight aperture and a post front sight. That system is far superior to the typical leaf/v-notch sights found on most service rifles and many MG's of the era. The BAR had sights calibrated for up to 1500 yards and the BREN has sight calibrated for up to 1800 yards, both of which were wishful thinking, really. At that range the firing is purely guessing at their effectiveness because the impacts can not be seen. So we've seen where the sights are capable and the cartridges are capable, but what about the shooter? Ultimately, the human eye can see only so far. The job of the light machine gun, or squad automatic weapon, is to directly support the men of the squad or platoon. That typically means direct lines of sight with fellow squad members and the enemy is the norm. If there is an enemy bunker that is 800 yards out and both the BREN and the BAR gunners have an opportunity to get in a position to comfortably deploy their bi-pod, I'm quite confident that both LMG's will be able to deliver sustained and accurate fire against the enemy while the riflemen advance. I would put the maximum effective range of both LMG's at 800 to 1000 yards depending on the gunner's experience and expertise.
What separates those two LMG's are other factors. The BREN had a 30 round magazine that fed from the top, and while in the LMG role (prone w/ bi-pod deployed) the BREN could be reloaded very quickly. The BAR was fed by a 20rd magazine from the underside, so while prone the gunner/loader had to shift the entire weapon to the side in order to clear the ground for a reload. Not as good as the BREN for sustained fire ability. The BREN had a inter-changeable barrel that was easy to swap out once it got hot from sustained fire. The barrel of the BAR was only able to be changed by a armorer with proper tools. The BREN was 25 lbs when loaded. The BAR was 19 lbs when loaded. In the LMG role, recoil was not a factor for either.
FWIW, next time you are out and about near a football field, stop and take a look at just how far 100 yards is (or the 140 yards between goalposts). Now think of how small an enemy soldier would be laying prone facing you with a rifle pointed at you at that range. Now thing of that same guy being 400 or even 800 yards away, behind a rock, tree, or even in a bunker. The human eye can only do so much and I cringe every time I see the History or Military Channel claim an exact 647 yard effective range on some particular rifle or gun, then claim something with identical abilities to have another 841 yards in effective range without explanation. There is a reason NATO forces use optics on their rifles on today's modern battlefield: it lengthens their effective range in a major way.