Learn English – What kind of construct is ‘would that that were all’

This use of would expresses in the past what will expresses in the present. It is typically used to report the past words or thoughts of someone in a story. In your example, the narrator would have said or thought at the time ‘there will be room for only a very few others’. Because the event as related is in the past, will be becomes would be. In contexts like this, would acts as the past tense of will.

(All) that much is a quantifier, but more importantly it's an idiomatic Negative Polarity Item.

That is, it can only occur felicitously within the scope of a negative trigger.
Like didn’t. Notice that you can certainly say

• I didn't understand all that much of what she said.

but you can't say

• *I understood all that much of what she said.

because there is no negative trigger present.

In the appropriate negative context, it means, basically, not much at all

• I doubt he enjoyed all that much of the play.
• It isn't as if she weighed all that much.

And it can also be used with other quantifiers and qualifiers besides much:

• She isn't all that gorgeous.
• If it's all that important, why don't we hear more about it?