In what ways have Buddhists responded to the apparent impossibility of saving all sentient beings? In what ways have Buddhists responded to the apparent impossibility of saving all sentient beings?

Do they claim that it isn't impossible, or only need be partially achieved, or that not all Buddha's took this vow, or what?
From the Mahayana viewpoint, the mind of enlightenment is the wish to achieve buddhahood in order to benefit all sentient beings. It is true we also often say "to set them in enlightenment" as well.

The omniscient mind of a buddha gives birth to the teachings of the buddha in that it causes him to teach us the Dharma. We say that the teachings of the buddha emanate from his omniscient mind (the wisdom truth body part of the dharmakaya). In a way, he teaches all of us but some do not have the ears to listen to him. This is merely an analogy. The meaning is that not all sentient beings have the karma to listen to the Buddha dharma but his enlightened deeds (in particular his teachings) pervade all sentient beings, just like the rays of the sun illuminate indiscriminately. However, the deeds of a buddha do not engage with all sentient beings, they are capable of doing so. 

In his Commentary to Maitreya/Asanga's Great Vehicle Treatise on the Sublime Continuum (mahayanottaratantra-ratnagotravibhanga) Gyaltsab Je says:


  These three can be illustrated: meditating on the path of utilizing
  the basic element results in the attainment of the dharmakaya of a
  perfect buddha whose enlightened deeds emanate and pervade to all
  sentient beings. These enlightened deeds of the dharmakaya are capable
  of engaging with sentient beings and exist exclusively for the mind
  streams of sentient beings. Because of this it is said that all
  sentient beings possess tathagata essence. This is similar in meaning
  to the Abhisamayalamkara when it says, "Because enlightened deeds are
  vast, it said that the buddha is all pervasive."


And


  Illustrating the teachings of conventional and ultimate truth with
  similes gives further explanation of the term "emanate" in the line,
  "The perfect buddha body emanates. " This is because "dharmakaya
  emanates" means "dharmakaya pervades," which in turn refers to being
  pervaded by enlightened deeds, and enlightened deeds engage with all
  sentient beings by way of the speech of the tathagatas teaching the
  dharma to various disciples. The capability of the enlightened deeds
  to engage with sentient beings is always present and this is the
  meaning of "pervaded by the dharmakaya.


Additionally, according to Mahayana tenets, the continuum of a buddha is not severed. His enlightened deeds will not stop as long as one being remains in samsara.