I think you guys are focused a bit too much on the strengths of the Alpha Legion and not enough on their...I hesitate to say weakness, but let's settle on limitations, all right?
First, it's time. They need time. Months at the least. Years...maybe decades or centuries to pull off their really big jobs. (Like the Swords of the Emperor).
If you need something taken out in a day, the Alpha Legion aren't your go to guys. Not saying they couldn't...but they definitely would not be as successful as say the Luna Wolves or World Eaters.
Define successful. If you need to minimize collateral/infrastructure damage, I'd posit that even the Luna Wolves would be among the least capable force at it, alongside the World Eaters, Iron Warriors, Space Wolves, and Blood Angels. Compared to say Salamanders (if dealing with a seccessionist force they were trying to bring into line, like the DA did in, I want to say, Unremembered Empire), Ultramarines, hell even Night Lord, if there was no cause for the terror tactics.
Now here's my conjecture, but I feel that the Alpha Legion's covert operations are more about being unseen as to the Raven Guard's invisible. If you can catch the nuance difference.
Now with a human or near-human civilization, that's very much possible. Even easy.
Something closer to the pre-fifth edition Necrons or the Tyranids, and I don't think you'll find them nearly as powerful. There's a lack of supply lines to take out and leadership is very decentralized. I'm not saying they wouldn't be able to target the Hive Tyrants or something, but not to any significant degree more than their brother Legions.
At the time of the design of the AL, or Alpharius himself, there was no such thing as the Tyranids or the Necrons to contend with, not to mention 17-19 other Primarchs and Legions - taking a nominal figure of 120K Legionaries in a Legion, that's roughly 2-2.25 million Astartes. In the vast scheme of things, on a galactic scale, it's not all that much, but in the scales we already have (where in 40K, there's arguably less than a million), not to mention the guidance of the Emperor, the Primarchs, and 10,000 years of enlightened advancement in a technological fashion, as well as fully operating forge worlds all designating one goal, the call of the AL to be james bond deep cover agents against the Nids and Necrons is questionable.
However, Extermination shows one thing - their pragmatic nature - they gave entire sectors cancer just to prevent one near exterminated alien life form coming back. If the Kryptman solution was on the table, then the Alpha Legion would be wholly for it - doubly so if it meant they could get some of the 500 Worlds of Rowboat with it.
Also, lastly, and perhaps most importantly, is their ability to work with their brother Legions. It seems they were disliked by the great majority of their fellow Astartes. This limits their own ability to wage war to what resources they can call upon. A major limitation.
This doesn't stop the Minotaurs in 40K. The AL have access to "better" or newer equipment - particular attention is paid to the Sicarans and the Mk6 Power Armour (the RG being the only other force with notable stocks of it), as well as possible insider information, and higher level clearances as shown by when they were the Ghost Legion (because lets face it, it's not going to be any other legion). More than even the TS, they believe in information is power. Despite the clunkiness of the writing and overall execution, the concept behind Deliverance Lost's AL sneaking aboard the RG shows how adept they are.
One of their biggest weaknesses is their hubris that comes from being the best. One of the old adages of Sun Tzu was never fight a battle you haven't already won. When fighting alongside the UM, they hindered the progression of a campaign to showcase how effectively they could win, just because they could. Their tendency to get caught in their own plots within plots could cause them to lose the focus necessary to push through and take that risk. Likewise, when their contingencies and plan b's all run out, they'll not have that safety net they're used to, leading to an early "bug out".
The comments regarding the speciality making them a big weak against non humanoids is also a bit of a null point. Yes, the Night Lords "Terror tactics" are all about spreading fear - but it's more than that, reading between the lines. It's about demolition of the psyche. Whereas the Alpha Legion will already have every battle tactic the enemy is going to make to hand, so they know where to avoid and where to strike, the Night Lords can force the enemy to attack to how they wish.
Think of it being the difference between having the blueprints to an enemy base, and being some kind of puppetmaster (AFB, but isn't that the title of Llansahai, the NL Apothecary?) directing the enemy. Whereas the RG are able to pick and pull and at a defence, forcing the enemy to expose their sweet spots by physical capabilities. I hope that's clear enough; it's how I interpret the differences between them. Alpha Legion are your Deep Cover Secret Agents (James Bond), the Special Forces/Green Berets are the Raven Guard, while the Night Lords are your CIA PsyOps teams.
To use an example from 40K, as much as I hate to do so, from Graham McNeils UM books. The NL's were attacking an UM position - they displayed (in typical McNeil fashion) just how chaotic and horrible and really quite not very nice people they were by splaying the ribs of captured UM's on their Rhinos. In McNeils writing, this serves nothing other than effectively an over exaggeration to make you feel nothing but hatred for the enemy, the equivalent of some king from prehistory invading another land on the pretext that they eat babies or something. But in reality, it's playing on the opponents psyche - makign them angry and doing irrational stuff.
In Unremembered Empire, Curze did exactly the same. He couldn't cause Fear, per se (although it became apparent he came close to it), but he could force them to dance to his tune. He knew for example that the UM would come to the aid of those still living, so he booby trapped heavily injured UM's, and when he was fighting the two Primarchs, he did it on prepared ground. When he wasn't able to do so, such as in Savage Weapons, he nearly lost in a more straight up fight against the Lion.
Against Nids, whose Hive Mind is such detached, they might not be able to fight to the best of their abilities using all of their specialised training, but like putting Navy SEALs in a regular battle order, you still have some highly effective infantry that you're going to miss out on.
Other legions would similarly suffer from losing their "shtick" or having their preferred style of warfare causing more problems.
Luna Wolves; their shock assaults would have less effect on the Nids. Sure, they have more of a linebreaker role, but that role comes with the certain expectation that the line will break. Unless they can cut through the numbers quick enough, then they're going to grind to a halt and lose all momentum.
World Eaters and Space Wolves; getting into handshake range with a nid is never fun. They might be better suited to killing everything, but there is also a psychological effect that comes from facing Berserker threats - that would be lost. Not to mention their significantly higher bodycount means only more Biomass extraction and an increased Nid threat later on.
Iron Warriors, Death Guard, Imperial Fists, Salamanders and Iron Hands; lets face it, these are all as stubborn as hell, and are unlikely to give ground; again, more biomass for the Nids.
Raven Guard; the Nids have no infrastructure to speak of, none of it is really truly essential. Their command structure is too decentralized to be too heavily effected, unless they can somehow strike at every Synpapse conduit.
Thousand Sons; Shadow in the Warp is a bit naughty for Psychic Powers.
Alpha Legion; they have no "blueprints" to exploit, short of preparatory tactics (Scorched Earth, etc).
Dark Angels; we don't know enough about them to say whether their tactics would suffer
Emperor's Children; being "perfect" might actually make them the best Nid counterers, outside of UM - they'd learn from their lessons. However, they'd also be a big risk, they'd likely lose a lot to learn those errors, which means more biomass.
White Scars; their speed could let them choose the battle, but they lose Psychological benefits of flanking etc
Blood Angels; possibly Berserker weaknesses ala SW/WE
Ultramarines; most likely to have some form of adaptive battle tactics. Only weaknesses is a certain pragmatism apparent in other legions, and apparently, Naivete; their lack of pragmatism could lead to failure of Kryptman style solutions, while their Naivete leads to gaps in defences (which Pollux pointed out in UE).
Word Bearers; again, hard to make comment over their battle tactics - they were the regular underachievers. The only thing which made them better was the Emperor's "intervention" on Monarchia, which then turned them to Chaos, ironically, so to keep them in a 40K "fanon" you'd have to discount the censuring of a legion, so they'd still be the lackadaisical proseletizers, and slower at reacting to the threats of the Nids. Conversely though, they may also have a greater battlefield presence - they're still battle capable, and can likely keep up the morale of humanoid allies if they can do their thing.
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