I don't get it it took decades to get an effective SAW chambered for the same round as the service rifle.
Well, that is, exactly, the rub of it.
The "light" machine gun concept was born during the second half of WW I. Although not truly recognized until after the second World War. The Light MG, morphed into (or merged; both arguments have merit) into a Squad-level MG.
That conflict also saw infantry units contract in size. Companies reduced to three Platoons, Squads compressed from ±15 to around 8. Which put a premium on single-man portable full auto fires.
Militaries are frugal things, so they all lumbered along with what they had, slowly, glacially, learning in brushfire and Cold War conflicts just how good--or bad--their training and doctrine really were.
What's "new" is a recognition that the Squad's base of automatic fires has to
exceed that of its rifles. While remaining single-man portable (meaning no dedicated Assistant Gunner, but that role is tasked to the rest of the Squad as needed). That rules out, mostly, in just moving a General Purpose MG out of the Company's Weapons Platoon down to the Squads; that, and a GPMG ties up about four troopies per MG, or near half a Squad.
Headcount in these things matters. When the BAR was developed, US Army used Rifle Companies with four Rifle platoons, and with Squads of 13 each. So, there was a lot of manpower to absorb the original concept of the BAR, which was 1 Gunner, 2 AG, and 2 Ammo bearers; five troops per BAR.
The world was simpler then, you fielded an army and you gave them ammo. One kind sufficed. And, the longer ranged the better. But, then, "intermediate" cartridges were discovered, for either practical or economic reasons (shorten your service rifle cartridge by a third, and you get a third more rounds for the same amount of powder).
This concept was dropped by brasshats almost as soon as it was implemented.
Yeah, this turns into book-length treatises, quickly.
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