Airborne infantry have Airborne Armor, meaning that on the move they have a .75 recieved accuracy modifier. Their M1 Carbines also have the smallest moving accuracy penalty, meaning that on the move the troops lose accuracy the least of all American weapons.
That said, the Carbines are also the weakest of the 3 common American rifles because of it's range modifiers and other factors. (BAR> Garand> Carbine)
Because of this, you are correct Airborne suck against infantry. Not sure why you're intending them to be used like that.
Airborne should NEVER, EVER be the main stay of your army. They should support troops for your Riflemen regiments or for your WSC units with Flame Engineer support, you need some sort of better anti-infantry dominance to be able to use the Airborne efficiently.
Use Airborne as dedicated AT squads, like Rangers. The two are similar in that they have large upkeep and cost (Which is why they should NEVER be your main army) but have well-to-do anti-tank weapons and are similar in that they have the Fire-up and grenade capabilities.
Their differences are that Airborne can be called in anywhere, and can be reinforced anywhere. While fighting behind enemy lines (IE You call them in and you lose a couple guys) if you noticeably win an engagement then you can reinforce on the spot. Their Satchels are nice at destroying or at least weakening bunkers for a final blow and if MGs are your issue you can use Fire-Up to flank around it quickly and lob a grenade in there to try and decrew it. Your other infantry, supposedly pinned, will be able to assist you in moments.
Now for RRs. I don't even know where you got that stupid "lol 10%" figure but I can assure you RRs are fine. Perhaps you should try to focus fire and stop your squad before firing as most heavier weapons are more accurate when still in COH. It's just the way it works.
Compare them to Bazookas, for instance. Both have similar-ish accuracies (RR Comes out on top IIRC), however Bazookas deal more damage while RRs deal much more penetration. This makes RRs very effective at dealing with tanks that have lower health and better armor. The only exception really is the Puma which has the phase-bug, but I don't even see it all too often anymore. It still has plenty good capability to kill the Puma or AC as long as you don't charge at it like a moron and use cover to your advantage.
Of course a freaking vet 3 flaming pio squad is going to kill your airborne. Vet 3 Pios have Elite armor, health regeneration, he was likely in cover, and you must not have been paying attention. IIRC Airborne armor is in the same as Elite armor and takes more damage from flames. Not to mention Airborne are not good against infantry as stated above.
If you see an AA Vehicle or emplacement (Flak 38, Whirlbel, Puma, Ostwind) Why the hell are you calling in airborne ANYWHERE near it?
They are helpless while dropping so those tanks can just fire away as they drop. If you visibly see weapons like this then call them in somewhere more convenient.
Airborne are just as useful as Rangers I feel. Neither can be used as a main force of an army primarily because of reinforce cost and upkeep, along with the fact that rangers are suspectible to snipers and fire, and airborne offer more tactical use but do not fare so well against infantry. Use BAR Riflemen to help support your airborne and use the airborne to lob grenades, decrew MGs in both buildings and bunkers, disrupt capping operations and offer decent counters to vehicles.
If you really want to preserve your airborne RR troops build a Triage Center
Airborne are weak unsupported as they are not viable against infantry. Build a triage center, and PAY ATTENTION to their health.
Keep them supported! Use BAR Rifles or combine arms help from WSC units and flame engineers.
Use the Fire-up and Satchels to your advantage to flank around weapons teams and make short work of enemy units garrisoned.
Call them in at strat points to help cut off enemy supply lines. They can be called in and reinforced anywhere.
Use their RRs to make short work of vehicles and tanks. Work well against most tanks except maybe the KT. Still a threat to Tigers. Sure they're expensive but if you're not looking to tech to 57mms or Hellcats it's what you have to do.
NEVER EVER CALL THEM IN NEAR AA WEAPONS.
Suggestion: Use Rifle Stickies to stop a vehicle cold and finish it off with RRs at range in cover. Works well against Pumas and vehicles of the like.