In WWII we issued rifle grenades to only one guy in the rifle squad. That's why one M1903 was retained per squad until the M7 GL could be developed for the M1 Garand. Once that was fielded in late WWII and all through Korea we had all M1/BAR equipped rifle squads, but still only one guy for the rifle grenades.
Late in Korea, the Army was going to go to a rifle squad structure that would have had two fire teams, each with a BAR and an M1 with GL, the rest armed with M1s, and one of them would have been a M1D giving one sniper per squad. Post-Korea there were funding and training issues, so the sniper idea fell by the wayside, to eventually be resurrected in Iraq.
But anyway in reality, the set-up was similar to what you have today with the M203. Only the M203 can be shot nearly twice as far, is twice as accurate, and you can shoot it twice as fast. Alot of people think everyone had rifle grenades, and now days we're somehow shortchanged, but the fact is that's not the case. We just went to a better way to delvier grenades by the same number of guys.
Alot of the reason not everybody carried rifle grenades was because back then you needed to use a grenade launching blank which would effectively take your rifle out of the picture as long as you were dicking with the rifle grenade. Today alot of that is eliminated by some designs that allow the use of ball ammo, but oddly enough using ball ammo doesn't throw the grenade as far, reducing range by some 50m. Your rifle is still tied up the entire time you are employing the grenade, so the squad is out that firepower during that time.
If you take a rifle squad today, there are already two guys with M249s, so they won't shoot grenades, and there's two M203's so there's no point there either. Dicking with a rifle grenade is pretty attention intensive, and it would be pretty debateable whether it would be a good idea for a squad leader or team leader to be fooling around with one rather than directing his troops. Better leadership will win a fight over the use of a rifle grenade most likely. If you have on guy with an M14 or other DMR, that leaves you with three or four guys that would be capable of effectively employing a rifle grenade and that would only be if your squad was full strength.
So the idea of where it would fit in today's squad is the big question. Since we only had one or two rifle grenade launchers per squad "back in the day", and they were replaced by the 40mm system that is much better, the question is not "why no rifle grenades?" but "why bring back rifle grenades?"
Special applications such as chemical delivery, illumination (which is still much better accomplished with the slap flare, and then the 40mm before a rifle grenade), less than lethal, and riot control would be some pretty valid applications it would seem. If you had a squad fire a couple dozen rifle grenades of CS at a mob, it would saturate the area pretty quick.
There have also been some interesting developments in disposible grenade launchers which would seem a better way to go than a rifle grenade if you wanted to arm everyone. Far lighter and less bulky than a LAW, and no flash/backblast. Not as much punch as a LAW, but as much as a 40mm.
Anti-tank use is pretty much out against anything built in the last thirty years, and a LAW or AT-4 is much better suited to AT use (the LAW replaced the old HEAT rifle grenade and M20 3.5" bazooka at the same time).
Add all of that to the expense in funds and time to train and field rifle grenades for minor return and it's just not worth doing.