M60s (and now M240s) are still issued, right along side the SAW. The saw didn't really "replace" the M60 per se, at least not in the infantry. It did change the way the GPMGs were issued, however, and for support units, it has replaced them.
Also, until the M60 came along, infantry units were lugging around 1919s, tripod and everything, and it was those that the M60 was intended to replace. When the M14E2 didn't pan out, though, they infantry never got its new automatic rifle. They would've been carrying the M60s anyway, just as how they carry both M60s/M240s and SAWs today. Each type of weapon has its own niche.
At any rate, I think full auto fire on a rifle is generally useless, regardless of whether it's magazine holds 20 or 30 rounds. The FAL is more controllable than the M14, I understand, but I can't imagine either being especially practical.
However, the BAR wasn't a traditional rifle. As I understand it, off the bipod full automatic rifle was controlable and could be done with some accuracy. And, heavy as the BAR was, I think it was still lighter than an M1919A6.
Still, many countries considered the FALO (heavy barrel FAL, used as an automatic rifle) to be an effective weapon, as they were used by the UK, South Africa, Australia, Canada, Israel, and countless others. So, you could count the number of designs on one hand, probably, but given the widespread use of the FAL line, one design was used by some 90 countries. If that's not sucess, I don't know what is.
The thing about the SAW is, a SAW type weapon HAS to be chambered in an intermediate catridge if it's to retain it's light weight. CAn they make a belt fed machine gun in a full house rifle round? Sure! But it ends up weighing more than the SAW, because 7.62x51mm requires a beefier weapon. If they could make a belt-fed .308 that weighed less than an M60, MAG58, or PKM, they would.
So, if your standard rifle fires the 7.62x51mm round, and you want automatic rifles to support your infantry squads and supplement your MAG-58-style machine guns, a BAR-type weapon (like the FALO) might just be the ticket.
A better idea, to my mind, though, would be a modernized, lightened Bren-type weapon, using the FAL gas system turned upside down the wtih magazine on top, allowing for longer, 30 round mags without getting in the way of prone firing. The British did this with the .308 caliber Bren gun, but I'm thinking of a modern, lighter-weight design, possibly firing from a closed bolt (is open bolt really necessary on a non belt-fed weapon?). The Bren did have a quick change barrel, and with the 30-round mags could really lay down the lead if necessary.
Just a thought. In reality, for those reasons the Bren was probably the better automatic rifle, but you've got to give the BAR credit, since it as developed almost 20 years earlier.