star wars - Why did Palpatine save Vader?

Really, why did he?

We see in the prequels (and even more so in Legends) how whenever a Sith becomes weak, he gets killed off by his own kind. One example is the Apprentice who becomes more powerful than the Master. Or as we can see in SWTOR trailers, Darth Malgus kills his badly injured master.

While the situation may be reversed here (it's the apprentice injured, not the master), we saw how Palpatine easily gave up Count Dooku simply because of this age, although he was a very powerful Sith.

And now it becomes clear that Darth Vader would no longer be able to become as powerful as the Emperor predicted because of his injuries (which indeed is the case: although foretold by a prophecy to be an extremely powerful Force user, losing his limbs cuts off of Vader's powers and abilities).

Doesn't it make a lot more sense for Palpatine to give up on Vader and find a new, more powerful apprentice?

Answer

Adding to DVKs point that an injured Vader might still be powerful:

While the situation may be reversed here (it's the apprentice injured, not the master), we saw how Palpatine easily gave up Count Dooku simply because of this age, although he was a very powerful Sith.

Injuries may be healed (even so with the advanced technology of the SW universe), but so far age cannot be reversed. So, helping Vader was an investment that could still pay off for many years. Dooku was just past his due date.

And now it becomes clear that Darth Vader would no longer be able to become as powerful as the Emperor predicted because of his injuries (which indeed is the case: although foretold by a prophecy to be an extremely powerful Force user, losing his limbs cuts off of Vader's powers and abilities).

Which might be indeed a good thing, given the work relationships Siths usually have. Anyway, Luke's prosthetic arm does not seem to cause him any trouble, so maybe you can assume that Vader's prosthetics do not weaken him much.

Doesn't it make a lot more sense for Palpatine to give up on Vader and find a new, more powerful apprentice?

Healing Vader appears to take a few weeks/months. If Palpatine had none ready, training a new Sith even to the rank of apprentice could take years.

Also, the Jedis were exterminated or in hiding, so even if Vader was weakened he was unlikely to be in the position of having to fight for his life with a lightsaber. Against non Jedis, Vader was almost as invulnerable as ever.

Note that in the original trilogy, Vader is more a representant of the Emperor than a true warrior (except in the final battle). Palpatine was holding his control of the Empire through the military, the Force was used to ensure that the military were kept in check. For this role, Vader's most important attribute were his "mind tricks" and his experience as a Jedi agent and officer.

Of course, these arguments could be countered by "but then Palpatine would have saved Dooku too". Against this, I would point that when Dooku was killed Palpatine still had Vader in reserve as a more valious agent, and the Rule of Two.