Last time I read it, German engineers already had the basic Cetme design mid war then fled to Spain after the war and kept working on it. The Spanish adopted it and the design eventually came full circle and was adopted in its actual country of origin.
To add to this, the German engineers who had pretty much already had the CETME completed, as an advanced STG44/45 didn't really flee to Spain as much as there just wasn't any work for them in Germany. Spain wanted to upgrade it's weaponry to that of the rest of the modern world and thus hired them down in Spain. There, they finalized the CETME as Spain would adopt it.
Interesting thing is that although the rifle was first developed in 7.92x33 (8mm Kurz) those engineers were thinking of adopting the 7.92x40 with the AL core bullet. While the German engineers weren't big on it, the Spanish wanted to retain something in 7mm. At the time, the .280 British was gaining favor. While all that was being discussed and progress halted, the U.S. came up with the 7.62x51 and politically mandated everyone wanting to belong to NATO must have it. (Read: shoved it up everyone's ass, then backed out on the FN deal). As this came about, most countries wanted to ally with us and started with the 7.62 in the FN-FAL. FWIW, the .300 Savage is the case the 7.62 is based on.
The Venezuelans wanted something in 7mm too as they liked the 7mmx57 they had in their bolt guns and FN-49's. When they adopted the FN-FAL, it was in 7x49mm (Liviano). The round was close to a .300 Savage necked down to 7mm. The .300 Savage necked down was also called the 7mm International here. Note: There is also a 7mm International Rimmed which is different. Neither got as far as SAAMI certification. , Although, I'm not sure that while SAAMI existed back int he 1950's, sporting compaines who founded it, wanted something different that they didn't have proprietary rights to. In 1957, the Venzuelans dropped the 7x49 in favor of joining the "U.S. Club" and adopted the 7.62. ( Of course, helping to sponsor the coup helped our chances of shoving the 7.62 up their asses too!)
All this said, while the FN was adopted by over 100 countries, the G-3/CETME was adopted by more than 70. Making it the second most formally adopted international rifle. Must be a reason. I also found the G3 (I shot them in Germany) didn't kick all that bad. Certainly no worse, than an M14. I've shot FAL's too and the recoil wasn't all that different to me. It punches like any .308 semi-auto (never fired full auto) then slams shut. If you want to get technical it only has two impulses vs. the three of the FAL. Neither was bad, IMO, and the M14 wasn't any worse. They kicked less than my .308 bolt gun.
As to "battle rifle" vs. "assault rifle" it really doesn't matter. The U.S. has adopted the M16 as the "Standard Service Rifle". Which means if you are going to battle, you will do battle with it, not an old "battle rifle". It's simply a progression of modern arms. It's like saying the M1 Garand wasn't a battle rifle, because it was semi-auto and everyone else had bolt guns. The fact that Hitler named the STG 44 the SturmGewehr doesn't make it a class all by it's own. It's just his propaganda tool to cover his fuck-up in not adopting it. Note: He thought they needed a bigger round too. The history of adoption of the M16 in the U.S. didn't follow the same route as the STG44. As noted it was initially meant to replace the M1 carbine. Then Gen LeMay got shown how much damage it could do to a watermelon. then SF got it through Thailand for VN. Suddenly, it was the main rifle. Fully transferred into our service rifle starting in 1967. So, it's our "battle rifle" now.