Yes a Holy Pally can definitely do this and I have with no problems apart from Itty dying lol.
I really think the problem with this is that the eagles aggro multiple targets, when I went a few times after Chistmas I would frequently see the Lock go down too fast, a Priest would die from nowhere, the Mage would follow suit. I actually felt sorry for Lumpy who would run all the way back from the grave only to last another minute before dying again. Consider there are 8 Eagles, a Warrior may get a little aggro on 4 but the Seeds will pull as at least their target as soon as it starts ticking, a Healer may pull another. When the aoe goes off it doesn't always hit everything as the eagles have moved, people are getting interrupted, healers have to heal different people and the next 8 can come down with 4 still up. Worse still, the Lock often pulls aggro on one of the elites.
So either all AoE and healers stand in the same spot or the Pally Tanks. Considering most of us pallies should have about 15K armor this is a good option.
Pally Tank
Buffs - Devotion, Righteous Fury (+30% threat), Kings
Consumables - Sup Wiz Oil (Aggro), Sporefish (20 stam 8 mp5), Maybe Draenic and Mageblood for mana
Other - Thorns, Ret aura if there are 2 pallies for increased aggro
Group - Put them in a group with any benefits you can, Lock Stam Buff, Shadow Priest Health/Mana regen, Shaman Totems
If you are an Engineer bring Stun bombs and Frost Grenades
Gear - Swap in any tanking gear you have that has Spell damage on it, I am lucky in this area as I have mostly epic tanking kit. I also throw in Darkmoon Card: Vengeance to try to gain a little more aggro and lovely Stam.
The Pally consecrates before they arrive so the birds travel into it. My was hitting around 150 damage each second but could have been higher. Have Seal of Righteousness up and just tab around the birds every few seconds. The Seal + Swing should do around 250 damage to a single target. Don't waste mana on Holy Shock unless you want to detonate a Seed.
The Lock should wait for Consecration and then spam Seed on the Eagles which will hit for about 1.6K each non crit.
The Mage should wait about 5 secs after consecration and spam Arcane Explosion which should hit for 650k ish. This should detonate the Seeds and hopefully 3 Seeds will be up by then. If that is not working then wait a little longer or only AoE when you see the Seed detonation as they will only have around 2K left at most to finish them off.
The Mage should stand in front of the Lock and if the Lock suddenly pulls a load of birds should Frost Nova.
Considering the eagles only have around 7.7K HP they should be down within 8 seconds (3 seeds should be enough + the other AoE damage)
The Pally should have HoTs on all the time.
When aggro is held and AoE is correctly executed this is easy mode, see below for the damage.
The problems arise when aggro is pulled on the Elites or when aoe is not fast enough and 2 sets of birds are out at the same time. I noticed that when I did it, I often got hit by the 2nd Add the Warrior was tanking or by the Amani Warriors that came from Behind. The Tanks have to be doing their best to gain as much aggro on both their targets and watching for loss of aggro. It may be an idea for the Tanks to go down the hill slightly (out of the consecration range) and then maybe pull them into it after 4 or 5 seconds.
22:51'58.140 Amani Eagle's Swing hits Lucklas for 280 Physical damage
22:51'58.468 Amani Eagle's Swing hits Lucklas for 281 Physical damage
22:51'58.484 Amani Eagle's Swing hits Lucklas for 305 Physical damage
22:51'59.890 Amani Eagle's Swing hits Lucklas for 305 Physical damage
22:52'00.093 Amani Eagle's Swing hits Lucklas for 286 Physical damage
22:52'00.343 Amani Eagle's Swing hits Lucklas for 226 Physical damage
22:52'00.468 Amani Eagle's Swing hits Lucklas for 246 Physical damage
22:52'00.468 Amani Eagle's Swing misses Lucklas
22:52'00.687 Amani Eagle's Talon hits Lucklas for 332 Physical damage
22:52'01.843 Amani Eagle's Swing dodged by Lucklas
22:52'01.859 Amani Eagle's Talon hits Lucklas for 266 Physical damage
22:52'02.109 Amani Eagle's Swing hits Lucklas for 244 Physical damage
22:52'02.453 Amani Eagle's Swing hits Lucklas for 292 Physical damage
22:52'02.453 Amani Eagle's Swing hits Lucklas for 264 Physical damage
22:52'04.078 Amani Eagle's Swing hits Lucklas for 118 Physical damage
22:52'04.500 Amani Eagle's Swing hits Lucklas for 230 Physical damage
22:52'04.500 Amani Eagle's Swing crits Lucklas for 494 Physical damage
22:52'06.062 Amani Eagle's Swing hits Lucklas for 237 Physical damage
22:52'06.625 Amani Eagle's Swing hits Lucklas for 241 Physical damage
22:52'06.625 Amani Eagle's Swing hits Lucklas for 237 Physical damage
22:52'08.078 Amani Eagle's Swing hits Lucklas for 127 Physical damage
22:52'08.468 Amani Eagle's Swing hits Lucklas for 238 Physical damage
22:52'08.968 Amani Eagle's Swing hits Lucklas for 242 Physical damage
22:52'10.203 Amani Eagle's Swing parried by Lucklas
22:52'12.156 Amani Eagle's Swing hits Lucklas for 219 Physical damage
22:52'16.750 Amani'shi Protector's Mortal Strike hits Lucklas for 2740 Physical damage
22:52'37.968 Amani Eagle's Swing hits Lucklas for 324 Physical damage
22:52'39.000 Amani Eagle's Swing parried by Lucklas
22:52'39.000 Amani Eagle's Swing misses Lucklas
22:52'39.921 Amani Eagle's Swing hits Lucklas for 305 Physical damage
22:52'40.453 Amani Eagle's Swing hits Lucklas for 337 Physical damage
22:52'40.453 Amani Eagle's Swing dodged by Lucklas