Did the Buddha intend for everyone to be enlightened? Was it the Buddha's intention that all beings to be free from suffering i.e. enlightened? 


  As the Dalai Lama acknowledges, “Buddhism has evolved differently
  in different times and places and yet the essential Dharma remains the
  same.  The Buddha’s prime concern was that all beings should find
  peace and freedom from suffering.


Did he say whether it was possible for everyone to be enlightened, so that there was no one left unenlightened? 
Would there then be no further (re)birth?
Answer to the original question:

The Buddha doesn't have wantings per se. He has uprooted craving. He simply helps beings to achieve freedom from suffering out of compassion. Beings are infinite, so any thought of making everyone enlightened is absurd. The Buddha certainly had no such ambition.

Answer to later additions:

I do not follow Dalai Lama and the use of the word 'concern' is not appropriate as enlightened beings are not 'concerned' per se. But there's a big difference between saying what everyone should do and having an ambition that everyone will do it. Ex: It's fine to say "everyone should eat healthy food at all times". Because that's just good advice to all. But having that as an ambition is not realistic.