You're describing an assault rifle.
As you say Browning (and the US Army) intended it as an 'assault rifle' (for lack of a better term), fired from the hip (hence the sling) using automatic fire. This didn't work. Quite probably the Pedersen would have worked better.
The BAR was later used as a LMG because of the USA resistance to the Lewis (which the USN and USMC, and Coast Guard, retained), for financial and other reasons. It was a pretty lousy support weapon, and the US never upgraded it, or replaced it despite the various attempts (T20, T24 et cetera).
And finally, as a military rifle, firing aimed semi-automatic shots from the shoulder, it didn't work because it was too heavy
Well, to their partial defense, nobody had the concept of an assault rifle at that time. (Nor, for that matter, anything like the modern LMG concept, but let's skip that.) They were, yes, struggling to discover the concept 25 years early, and not doing it perfectly the first time, but that's the nature of the beast.
Additionally, you have to just understand the context. They were ALREADY in a war, and the only other thing they had for the role RIGHT NOW was the Chauchat. And, oh dear, the Chauchat sucked royally.
Even the French 8mm Lebel cartridge version suffered from jamming in dirty conditions (like, you know, the mud in WW1

), low fire rate, piss-poor accuracy (the long recoil design -- i.e., the whole barrel would slide back, not just the bolt -- made it shake like all hell), and a host of other problems.
But even that was perfect compared to the .30-06 version for the Americans. That sucked more ass than the vacuum toilets on the ISS. You'd be lucky if it could even extract more than none when firing. It was universally HATED by the US troops. And generally, all you need to know is that it's still considered the worst automatic weapon ever designed.
Could they have stopped, tried a bunch of other weapons, and chosen the best one? Technically yes, but they didn't have the time for it. They needed an automatic rifle, like, LAST YEAR.
Could they choose or design better with today's knowledge? Undoubtedly. But they didn't have that knowledge at the time.