Did the Buddha intend for everyone to be enlightened?
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.