This answer says that (with the wrist intact) they'd do no better. Actually, not only would they do no better but they'd do worse, for two reasons: reduced swordsmanship, and inability to do other useful things with their non-dominant hand, like use a shield.
Effective use of a sword does not just rely on swinging the arm. The wrist is critical for first-order refinement (aiming, feints, off-side strikes, and so on), and the fingers and thumb are useful for second-order refinement. The latter depends some on the type of gauntlets being worn, but even "mitten-style" ones that don't allow good use of the fingers still afford the thumb movement independent of the fingers. This matters.
To see this, try holding something the size and weight of a sword (a wooden dowel will do). Grip it tightly in your fist and swing; observe how it moves. Now grip it loosely and swing; observe what the "sword" is trying to do. Now adjust your grip as you swing and observe that. (I'm not describing good technique here; the point of this is to show that the option space is larger if you can use your fingers.)
Further, you describe this as being done to the non-dominant hand, presumably because you want the dominant hand to remain available for other things. So, either your elite troops are fighting with two swords or they're using their weapon in the lesser hand. If they've already been trained to use a sword in the dominant hand, they'll have to relearn some of what they know. It's not just about the hand; when you switch to fighting with the "off" hand you have to adjust your stance, your power, your targeting, and your defense. (And you'll still never be as good with that hand, unless you're truly ambidextrous.) Your "elite" troops will be less-effective swordsmen than those same troops using swords in their dominant hands. If the plan is for them to use weapons in one hand they'll be less skilled; if the plan is for them to use weapons in both hands (two-sword fighting) then you've taken away the shield.
And that brings me to the second point: in battle formations (where battles are fought in open fields, as was often done in the middle ages), where troops fight as a unit, the shield wall is very important. It presents better defense when being attacked from the front, and in some configurations it protects from archery fire. Your elite troops will not be able to participate in the shield wall -- or, if they do, their main contribution will be holding shields, as we've already discussed their reduced offensive abilities.
There's more to battles than units formed up in shield walls, true -- your elite troops might be used as two-weapon harriers, who operate independently and try to disrupt enemy lines. Their mortality rate will be high (it's easier to pick off one guy acting alone than someone who can't easily be flanked or attacked from behind), but hey, they're supposed to be super-loyal, right? But they'll also be at a distinct disadvantage if they have to, say, crawl through underbrush, climb trees to drop on enemies from above, or do anything else where having both hands would help them get into position.
Also, as pointed out in this answer and in comments, the consequences of a broken or stuck weapon are much, much more severe in this scenario, as are the consequences of injury to the dominant hand.
And on top of all that, as others have noted, you're impeding their ability to do the tasks of daily living (like getting dressed) and ruining them for non-military tasks completely -- maybe ok if you're planning for them all to die in battle, but not very versatile.