Now I see what you mean, you are confusing two spaces. Imagine a finite two dimensional space, say one meter in each dimension. You are now almost certainly picturing something like a sheet of paper one by one meter floating in our three dimensional space in front of you or maybe the surface of a table of that size. The important thing is that this is not a picture of a finite two dimensional space, it is the picture of a finite two dimensional space embedded in a three dimensional space.
If you would crumble up the sheet of paper to a small ball than that would be a different embedding of the two dimensional space in our three dimensional space but importantly the two dimensional space would remain unchanged. That is a two dimensional creature living on the sheet of paper could not tell the difference between living in a nicely flat sheet of paper and living in the crumbled up ball of paper.
And that is the same in your picture, space is the string between A and B and you can not tell the difference between it being embedded in a straight or curled up manner. Even more importantly a space does not require any embedding at all. It is a limitation of our brains that we can only imagine a two dimensional space as a sheet embedded in our three dimensional space. A and B can get further apart without being embedded in any space and moving away from each other in the embedding space or having new space curl up in between them.
So your mistake is to assume that our space must be embedded in another space in order to be able to expand. This might be the case but it is not necessary and we have no evidence for that. It also starts an infinite regress, why wouldn't the space our space is embedded in be embedded in yet another space? And then that one and the next one and so on.
If our space was indeed embedded in another space and curled up there, than your suggested faster trip from A to B would be the equivalent of a two dimensional creature leaving the crumbled up ball of paper into the third dimension, moving a bit through our three dimensional space, and then falling back into the paper at a point nearby in our three dimensional space but potentially far away from the starting point if you moved only inside the crumbled up sheet of paper. You would literally have to leave space-time and move through a space that is not our space-time.
This is in some sense very similar to wormholes but in that case you do not have to leave space-time and move freely through the embedding space but you can follow shortcuts inside space-time due to the topology of it. But again there is no need for any embedding space even if all the pictures of wormholes show space-time with the wormhole embedded in another space.