[...] whether it packs away nicely into a bounded area [...]
That is nothing you should worry about, with enough deformation you can make a map of any shape you like for any space. You must not conclude from looking at a map of the Earth using for example a Mercator projection that there are somewhere four straight edges meeting at right angles and where you can just fall off Earth. You also must not naively make conclusions about the relative sizes of things as they get heavily distorted. The same holds for this map of hyperbolic space, it is really misleading - in the space there is no boundary, there is no special point in the middle, it does not pack nicely into a circle in any meaningful way, ... Those are all just artifacts of the way this map is drawn and it has nothing to do at all with the underlying space.
That is nothing you should worry about, with enough deformation you can make a map of any shape you like for any space. You must not conclude from looking at a map of the Earth using for example a Mercator projection that there are somewhere four straight edges meeting at right angles and where you can just fall off Earth. You also must not naively make conclusions about the relative sizes of things as they get heavily distorted. The same holds for this map of hyperbolic space, it is really misleading - in the space there is no boundary, there is no special point in the middle, it does not pack nicely into a circle in any meaningful way, ... Those are all just artifacts of the way this map is drawn and it has nothing to do at all with the underlying space.